A Game-Theoretic Foundation for the Wilson
Equilibrium in Markets with Adverse Selection∗
Wanda Mimra† and Achim Wambach‡
July 2016
Abstract
In competitive common value adverse selection markets, existence of a purestrategy equilibrium yielding a second-best efficient allocation is often justified
by appealing to Wilson (1977)’s concept of ‘anticipatory equilibrium’. The
‘anticipatory equilibrium’ is based on the notion that all market participants
expect unprofitable contracts to be withdrawn, however, it has so far not been
well-founded game-theoretically. We extend the seminal Rothschild and Stiglitz
(1976) model on competitive insurance markets with asymmetric information
in the spirit of the ‘anticipatory equilibrium’ by introducing an additional stage
in which initial contracts can be withdrawn repeatedly after observation of competitors’ contract offers. We show that an equilibrium always exists where consumers obtain their respective second-best efficient Miyazaki–Wilson–Spence
(MWS) contract. This requires latent contracts on offer. However, other allocations, such as jointly profit-making contracts can also be sustained as equilibrium allocations. We further allow for contract addition as in Riley (1979)’s
‘reactive equilibrium’. Allowing for contract addition does not change the set
of possible outcomes.
JEL classification: C72, D82, G22, L10.
Keywords: asymmetric information, competitive insurance market, contract withdrawal.
∗
We would like to thank Keith Crocker, Vitali Gretschko, Christian Hellwig, Tore Nilssen, Philip
Reny, Andreas Richter, Klaus Ritzberger, seminar and conference participants at Bonn, Frankfurt,
Köln and Toulouse for very helpful comments and discussions.
†
Center of Econonic Research, ETH Zurich, Zuerichbergstrasse 18, 8092 Zuerich, Switzerland.
Email: [email protected]
‡
University of Cologne, Department of Economics, Albertus Magnus Platz, 50923 Cologne, Germany. Email: [email protected]
1
1
Introduction
The Rothschild and Stiglitz (1976) model on competitive insurance markets with
adverse selection is widely considered as one of the seminal works on asymmetric
information common value markets besides Akerlof (1970). Yet, it entails a puzzle:
an equilibrium in pure strategies may fail to exist altogether. To be precise, Rothschild and Stiglitz (1976) show in a simple screening game in which insurers offer
contracts first and then consumers choose, that if the share of high risk types is low,
the candidate separating, separately zero-profit making Rothschild-Stiglitz (RS) contracts cannot be tendered in equilibrium as they might be overturned by a pooling
contract. However, pooling cannot be an equilibrium as insurers would then have an
incentive to cream skim low risks.
This potential non-existence of equilibrium has received much attention ever since;
subsequent research has addressed the non-existence problem by considering mixed
strategies (Rosenthal and Weiss 1984; Dasgupta and Maskin 1986; Farinha Luz 2015),
introducing equilibrium concepts that differ from Nash-equilibrium (Wilson 1977; Riley 1979), extending the dynamic structure of the game (Jaynes 1978; Hellwig 1987;
Engers and Fernandez 1987; Asheim and Nilssen 1996; Netzer and Scheuer 2014;
Diasokos and Koufopoulos 2015) or modifying assumptions about insurer or contract
characteristics (Inderst and Wambach 2001; Faynzilberg 2006; Picard 2014; Mimra
and Wambach 2015).1,
2
To date, due to its intuitive appeal, one of the most referred to solutions is still the
Wilson (1977) equilibrium.3 Wilson (1977) considers the RS game structure, but introduces the ‘anticipatory equilibrium’ concept: In this concept, an expectation rule
is imposed such that “each firm assumes that any policy will be immediately withdrawn which becomes unprofitable after that firm makes its own policy offer” (Wilson,
1977, p.1). Wilson shows that the anticipatory equilibrium concept leads to a pooling
equilibrium in which low risk utility is maximized subject to a zero-profit condition.
1
There are yet methodically different strands in the literature. Ania, Tröger, and Wambach
(2002) take an evolutionary game theory approach, Guerrieri, Shimer, and Wright (2010) consider
a competitive search model; other directions use cooperative concepts (Lacker and Weinberg, 1999)
or a general-equilibrium framework (see e.g. Dubey and Geanakoplos (2002) or Bisin and Gottardi
(2006)). For a recent overview see Mimra and Wambach (2014).
2
This literature assumes exclusive contracting. Attar, Mariotti, and Salanié (2014) and Ales
and Maziero (2013) consider non-exclusive contracting in RS environments. In both models, a pure
strategy equilibrium may fail to exist. The key insight in these models of non-exclusive contracting
is that in equilibrium, if it exists, contracts are linear and offer positive insurance only for the high
risk type.
3
For recent work using the Wilson equilibrium notion for definition of equilibrium in the labor
market, see e.g. Stantcheva (2014).
2
Extending the analysis to contract menus, Miyazaki (1977) and Spence (1978) show
that the anticipatory equilibrium concept results in an allocation with separating,
cross-subsidizing, jointly zero-profit making contracts that are second-best efficient,
the famous Miyazaki–Wilson–Spence (MWS) contracts. However, surprisingly, despite its appeal the logic behind Wilson’s anticipatory equilibrium concept lacks a
sound game-theoretic foundation to date.4
The contribution of the present paper is threefold: First, we propose a game-theoretic
model of contract withdrawal that spells out the strategic interaction behind the anticipatory equilibrium’s logic. Second, we show that in our model of contract withdrawal, the MWS allocation can be sustained as equilibrium allocation. However,
this results needs to be qualified: It requires latent contracts, i.e. contracts on offer
that will not be taken out. Furthermore, many more allocations can be sustained
as equilibrium allocations. We show however that the MWS allocation is the only
allocation that can be sustained without withdrawal on the equilibrium path. Last,
we augment the model by also allowing contract addition. This allows us to confirm Riley’s (2001) conjecture that in a game where firms are allowed to either add or
drop offers, both the MWS and the RS allocation can be sustained as Nash equilibria.
In the game we propose, we introduce an additional stage into the RS model in which
firms can withdraw contracts (repeatedly) before consumers make their choice, but
after observing the contract offers of competitors. Importantly, the withdrawal stage
ends endogenously. We confirm the anticipatory equilibrium’s logic that the possibility of contract withdrawal prevents cream-skimming deviations that upset the
MWS contracts in the original RS set-up. However, to sustain the MWS allocation,
not only the MWS contracts, but also a continuum of contracts on the low risk fair
insurance line as well as the RS contracts have to be on offer as latent contracts. The
reason is that the explicit possibility to withdraw contracts allows for sophisticated
deviating strategies that are prevented by latent contracts that, off the equilibrium
path, attract low risks away from such possible deviations.
The possibility to retract contracts however also provides firms with adequate threat
points to sustain allocations different from the MWS allocation, such as profit-making
allocations. More generally, (individual) contract withdrawal leads to a multiplicity
4
Some, in particular more recent research on the equilibrium inexistence problem yields the MWS
allocation (Asheim and Nilssen 1996, Picard 2014 and Mimra and Wambach 2015), however, the
economics in these models is quite different. Citanna and Siconolfi (2015) study a Walrasian market
for mechanisms in large insurance economies and adapt Wilson’s (1977) notion of ’anticipatory
equilibrium’ to define a competitive equilibrium.
3
of equilibrium allocations. We show however that the MWS allocation is the only
equilibrium allocation without contract withdrawal on the equilibrium path. This
feature has strong appeal in real (insurance) markets: One interpretation is that
MWS contracts are a sustainable outcome, offered and traded over time, as long as
no other contracts are offered. All other equilibrium allocations are not amenable
to such an interpretation: In the corresponding equilibria firms regularly withdraw
contracts, and transactions do only take place after rounds of withdrawals, which is
hard to reconcile with real world markets.
Related Literature
To attain existence of a pure strategy equilibrium in adverse selection markets with
common values, a small literature has considered contract withdrawal. An early contribution is Hellwig (1987), which differs in the timing of the game: firms make single
contract offers and may decline to fulfill the contract and thus exit the market in
a third stage after consumers have already chosen their insurance contract in stage
2. Hellwig (1987) argues that the Wilson pooling contract corresponds to a stable
equilibrium of this three-stage game. However, note that when allowing for contract
menus instead of single contract offers and individual contract withdrawal, the MWS
contracts do not constitute equilibrium contracts in Hellwig’s game as any firm would
have an incentive to withdraw the loss-making high risk contract.5
In contrast to Hellwig (1987), Netzer and Scheuer (2014) consider contract withdrawal
before consumers choose. However, as in Hellwig (1987) contract withdrawal is restricted to complete market exit, i.e. all contracts of the firm are withdrawn or none.
This does not allow for the flexibility of firms’ strategies envisioned by Wilson: When
allowing for contracts withdrawal, firms’ strategy space should not be restricted to
only withdraw the complete set of offered contracts, but instead allow (optimal) withdrawal of individual contracts. Our model considers this more general framework and
shows that the second-best efficient MWS allocation can be attained, however this
requires latent contracts as part of equilibrium characterization. Furthermore, we
additionally incorporate Riley’s ‘reactive equilibrium’ notion by allowing for addition
of contracts in the second stage. Contract addition has been analysed by Engers and
Fernandez (1987). Engers and Fernandez (1987) provide a game-theoretic foundation
for Riley’s reactive equilibrium by considering a game where insurers add contracts
5
In recent work, Diasokos and Koufopoulos (2015) reconsider Hellwig’s timing of the game, but
allow for contract menus and endogenous public commitment on menus. Firms are however only
allowed to withdraw the complete menu, but not individual contracts.
4
repeatedly, and consumers choose contracts after the final contract addition has been
made.6 We incorporate the addition of contracts into our Wilson game. This allows
us to shed light on Riley’s (2001) conjecture that without further selection criteria,
both the RS and the MWS allocation are equilibria of a generalized game with contract withdrawal and addition.
Endogenously ending contract withdrawal (and addition) in our game allows for a
rich strategic interaction of firms. Through this strategic interaction, the set of equilibrium allocations is large, which mirrors recent folk theorems in the competing
mechanism literature. Szentes (2015) for instance provides a folk theorem when contracts are cross- and self-referential. As all folk theorems, they inform us more about
extreme theoretical cases and the feasibility frontier with a rich contract space rather
than real-life situations. We bridge the gap between such an approach and the simpler contracts studied in most adverse selection applications by looking at realistic
contracts and plausible strategic interactions. Thus, we complement this literature
by showing that similar results are obtained when the contract space is ordinary and
contingent contracting takes the simple form of the possibility of contract withdrawal
upon observing competitors’ contract offers.
The rest of the paper is organized as follows: In the next section, the game is introduced. Section 3 provides an existence result for the MWS equilibrium, and section 4
provides a full characterization of equilibrium allocations. In Section 5, the analysis
is extended to allow for the addition of contracts.
2
The Wilson game
There is a continuum of individuals with mass 1. Each individual faces two possible
states of nature: In state 1, no loss occurs and the endowment is w01 , in state 2 a
loss occurs and the endowment is w02 with w01 > w02 > 0. There are two types of
individuals, an individual may be a high risk type (H) with loss probability pH , or
a low-risk type (L) with loss probability pL , with 0 < pL < pH < 1. Insurance is
provided by firms in the set F := {1, ..., f, ..., n} with n ≥ 5. Firms do not know, ex
ante, any individual’s type. If an individual buys insurance, then the initial endowment ω0 = (w01 , w02 ) is traded for another state-contingent endowment ω = (w1 , w2 );
we say the individual buys insurance contract ω. The set of feasible contracts, Ω, is
6
In Engers and Fernandez (1987) insurers choose contracts sequentially rather than simultaneously, the order of moves being exogenously given.
5
given by Ω := {(w1 , w2) ∈ R2 }. Preferences satisfy the von Neumann-Morgenstern
axioms, expected utility of a J-type individual, J ∈ {H, L} from chosing a contract
ω ∈ Ω is abbreviated by uJ (ω) := (1 − pJ )v(w1) + pJ v(w2 ) where v is a strictly
increasing, twice continuously differentiable and strictly concave utility index.
The timing of the game is as follows: First, firms set contracts simultaneously and
observe their competitors’ contract offers. Then, firms can withdraw contracts potentially repeatedly for several rounds whereby firms observe their competitors remaining
contract offers after each round. Contract withdrawal is possible as long as at least
one contract was withdrawn by any firm in the previous round. After contract withdrawal ends, consumers make their contract choice. Formally, the game proceeds as
follows:
Stage 0: The risk type of each individual is chosen by nature. Each individual
has a chance of γ, 0 < γ < 1 to be a H-type, and of (1 − γ) to be a L-type.
Stage 1: Each firm f ∈ F offers an initial set of contracts Ωf0 ⊂ Ω. The offered
sets are observed by all firms before the beginning of the next stage.
Stage 2: Stage 2 consists of t = 1, 2, ... rounds. In each round t, each firm f ∈ F
can withdraw a set from its remaining contracts. After each round, firms observe
the remaining contract offers of all firms. Denote by Ωft firm f ’s contract set on
offer at the end of round t; Ωft ⊆ Ωft−1 . If, at some round t, Ωft = Ωft−1 for all
f ∈ F , this stage ends. Denote the final round in stage 2 by t̂.
Stage 3: Individuals choose among the remaining contracts
S
f ∈F
uninsured.
Ωft̂ or remain
For mathematical convenience we permit only strategy profiles such that for each firm
the initial set of contracts offered in stage 1 and each firm’s contract set on offer after
each round of withdrawal is compact, i.e. for all f ∈ F and t, Ωf0 and Ωft are compact.
To incorporate no insurance in a simple manner, we impose that (w01 , w02 ) ∈ Ωft for
all f ∈ F and t ≥ 0.
Before proceeding, let us discuss the difference of our setup to RS and how this implements the Wilson concept: The RS game corresponds to stages 0, 1 and 3. In
6
this reduced game, a pooling contract or more generally cross-subsidizing contracts
cannot be sustained as equilibrium contracts as insurers would always try to cream
skim low risks. In Wilson’s ‘anticipatory equilibrium’ concept, such cream skimming
deviations are not profitable because the expectation rule is that cross-subsidized
contracts at non-deviating insurers would be withdrawn since they become unprofitable after introduction of the cream-skimming contract, rendering the deviation
unprofitable. We implement this concept by adding stage 2. However, when instead
of imposing an expectation rule, firms are explicitly allowed to withdraw contracts
after observation of competitor’s contract offers, for the Wilson reasoning to hold in
a game with contract menus, contract withdrawal has to end endogenously as in our
model specification: with a fixed number of withdrawal rounds, a single firm would
always be able to profitably deviate by withdrawing a cross-subsidized contract in
the last round.
Let ∆t :=
S
f ∈F
Ωft , Ωt := (Ω1t , ..., Ωnt ) and let ht = (Ω0 , Ω1 , ..., Ωt−1 ) denote the history
in the beginning of round t. The set of all histories in round t is denoted by Ht .
The game does not entail any strategic role for consumers. In any equilibrium,
consumers will buy the best contract available at stage 3. We denote by ω̄tJ =
J
J
) the contract such that
, w̄2,t
(w̄1,t
ω̄tJ ∈ arg max uJ (ω)
ω∈∆t
and
J
J
w̄2,t
≥ w̃2,t
∀ ω̃tJ ∈ arg max uJ (ω)
ω∈∆t
J
J
J
Let
n k̄ (Ωt ) denoteo the number of firms offering ω̄t at the end of t, i.e. K̄ (Ωt ) :=
f ∈ F | ω̄tJ ∈ Ωft and k̄ J (Ωt ) := |K̄ J (Ωt )|. We assume throughout that the strategy
of a consumer of type J is to choose ω̄t̂J at firm f ∈ K̄ J (Ωt̂ ) with probability 1/k̄ J (Ωt̂ ).
That is, consumers buy the contract which gives them the largest utility. In case of
indifference, they buy the contract with the largest indemnity. This tie breaking
rule is part of the equilibrium strategy of the consumers. It allows that in case of a
tie, a high risk buys the contract with the larger indemnity, as she does in the RS
equilibrium or in the MWS equilibrium. As the set of contracts is compact, a solution
to the maximization problem of the consumer exists. With this specification of the
consumer strategy, we can insert consumer choice directly into the firms’ objective
function such that the game reduces to a game of complete information between
7
firms.7
We denote the profit of firm f when stage 2 ends after round t̂ by
H
H
H
H
π f (Ωt̂ ) := γ[(1 − pH )(w01 − w̄1,
t̂ ) + p (w02 − w̄2,t̂ )](1/k̄ (Ωt̂ ))1I{ω̄ H ∈ Ωf } +
t̂
L
(1 − γ)[(1 − p )(w01 −
L
w̄1,
t̂ )
L
+ p (w02 −
t̂
L
L
w̄2,
t̂ )](1/k̄ (Ωt̂ ))1I{ω̄t̂L ∈ Ωft̂ }
(1)
where 1I is an indicator function. As we did not restrict the sets of contract offers in
stage 1 to be finite, stage 2 does not necessarily end. For t → +∞, we specify that
firms make zero profits.8 A strategy for a firm f is a sequence of maps {αtf }∞
t=0 where
each αtf maps histories in sets of contracts, i.e.
αtf : Ht → P(Ω)
ht 7→ Ωft
Ωft compact for all t ≥ 0 and the restriction Ωft ⊆ Ωft−1 for all t > 0. The game is
a (potentially) infinite horizon multi-round game with observed actions with payoffs
bounded from above. In the following, we analyze subgame perfect Nash equilibria
of the game.9 We say that a contract menu (ω H , ω L ) constitutes an equilibrium
allocation if an equilibrium of the game exists where consumers of type H (L) buy
contract ω H (ω L ) in stage 3.
3
Equilibrium with MWS allocation
The MWS contracts
Let us first recall the Miyazaki-Wilson-Spence (MWS) contracts, which of all separating contracts that jointly break-even are the pair that is most preferred by the
7
We consider a specific strategy of the consumer, where the tie breaking rule is not history
dependent. Propositions 1 and 2 are existence result, such that this assumption on the strategy of
the consumers is without loss of generality.
8
Let us stress that it is solely out of convenience that we do not restrict the set of feasible
contracts Ω and hence do not assume contract offers to be finite such that stage 2 does not necessarily
end. As will become clear below, our main result remains to hold if we would consider a discrete
contract grid and thus a finite number of stage 1 contract offers and stage 2 contract withdrawals.
Only for some very specific equilibria in section 3 we make use of the assumption that the contract
space is continuous.
9
There of course exist trivial equilibria with infinite withdrawal by some firm in which firms
make zero profits and no contract choice of customers. We will discard these in the subsequent
analysis.
8
H
L
low risk type. The MWS contracts, which we denote by ωM
W S and ωM W S , can be
obtained as the unique solution10 to the following maximization problem:
max uL (ω L )
ω L ,ω H
(2)
s.t.
uH (ω H ) ≥ uH (ω L )
(3)
H
uH (ω H ) ≥ uH (ωRS
)
(4)
γ[(1 − pH )(w01 − w1H ) + pH (w02 − w2H )]+
(1 − γ)[(1 − pL )(w01 − w1L ) + pL (w02 − w2L )] ≥ 0 (5)
H
where ωRS
= (w01 − pH (w01 − w02 ), w01 − pH (w01 − w02 )) is the fair full insurance con-
tract for the high risk type. Note that the MWS contracts are second-best efficient.11
H
L
H
Denote by ωRS
and ωRS
the H-type and L-type RS contracts; ωRS
specifies full
coverage while the expected zero-profit condition for insurers on this contract holds
L
while ωRS
is pinned down by maximizing uL (ω) subject to the expected zero-profit
condition for the insurer on this contract and the H-type incentive compatibility
constraint, which both become binding. Note that the MWS contracts correspond to
the RS contracts when (4) is binding. When (4) is not binding, MWS contracts are
such that the fully-insured H-types are subsidized by the partially insured L-types.
We will focus on this more interesting case for the remainder of this paper.12
The MWS and RS contracts are shown in Figure 1. The axes in Figure 1 depict wealth
in state 1 and 2 respectively, and point (w01 , w02 ) shows the initial endowment. The
solid lines are the fair insurance, i.e. zero profit lines for each type, where the flatter
one corresponds to the fair insurance line for H-types. The dashed line is the pooling
zero profit line. The curved lines denoted by uL and uH are indifference curves of the
corresponding type. The dotted curve gives all L-type contracts that jointly with a
corresponding incentive compatible full insurance H-type contract yield zero profits
overall.
Preliminaries: Potential deviations and their economic implications
Below we show that there exists an equilibrium where on the equilibrium path the
MWS contracts are traded. Before proceeding to the formal analysis, and in order
10
See e.g. Miyazaki (1977).
This was shown by Crocker and Snow (1985).
12
This is precisely when equilibrium fails to exist in the RS set-up when firms are allowed to offer
contract menues. Our results hold as well for the case that the MWS contracts correspond to the
RS contracts.
11
9
w2
H
ωM
WS
H
ωRS
L
ωM
WS
uL
uH
uH
L
ωRS
w02
w01 w1
Figure 1: MWS and RS contracts
to get some intuition for the reasoning in the formal analysis, we first discuss how
potential profitable deviations from the equilibrium path might look like, and what
their economic implications are:
Stage 1: Contract setting
At this stage, firms set contracts. If all firms are supposed to offer the MWS contracts, then a potential deviation might be the attempt to cream skim, i.e. to attract
the low risks. In the next section, where we discuss other allocations such as profit
making allocations, a potential profitable deviation might also be to undercut the
competitors. This latter deviation strategy is just a standard price competition effect, while the former deviation strategy is at the heart of the analysis by RS and
subsequent work.
Stage 2: Contract withdrawal
Once contracts are set, firms can only react by withdrawing contracts. In case the
MWS are the equilibrium contracts, on the equilibrium path there will be no contract
withdrawal. This is different for the case of other equilibrium allocations, where withdrawal on the equilibrium path is necessary. Potential deviations from this equilibrium behavior might take one out of three forms: (i) In the case of a cross subsidizing
equilibrium, firms might be tempted to withdraw loss making contracts. This case is
of relevance from a practical point of view (why should an insurer keep on offering
loss making contracts?), but has received less attention in the academic literature.
10
(ii) In the case of an equilibrium allocation with profit making contracts, a potential
deviation might take the form of withdrawing one of the profit making contracts.
Suppose firms offer two separate contracts for the high and low risks, where both
contracts individually would be profit-making. Now it can be the case that if the low
risk contract is removed, low risks will buy the high risk contract instead, and the
profit per low risk customer increases. So if there is a single insurer offering these
two contracts, it would be optimal for this insurer to withdraw the low risk contract.
If there are two firms, and if one of them removes the low risk contract, it might be
optimal for the second one to follow. This will be the case if the profit per low risk
customer at the original contract is less than half the profit per low risk customer
at the high risk contract. Thus even if several firms are offering contracts, withdrawal of profit making contracts might be a profitable deviation. This withdrawal
of profitable contracts has not yet been considered in the academic literature. (iii)
A deviating insurer might enforce contract withdrawal by the other firms. For example, an insurer might offer a cream skimming contract, then wait until the others
react to this by withdrawing their contracts which would otherwise make a loss, and
then withdraw the initial cream skimming offer himself. That might be profitable if
another contract, which is also offered by this deviator, would then be chosen by the
policyholders. We call this deviation strategy forcing of withdrawals. This forcing of
withdrawals is a direct consequence of allowing for the ability to withdraw contracts,
as proposed by Wilson, but has not been considered in the literature yet.
Next we discuss why these deviations are not profitable with the equilibrium strategies we will propose:
Contract withdrawal is used to prevent deviating behavior, be it at the contract
setting stage or at the withdrawal stage. Contract withdrawal entails the following
elements: To prevent cream skimming, contracts which would make a loss will be
removed. Thus cream skimming is not profitable. Similarly, if a firm withdraws the
contract designed for the high risks, i.e. if the firm withdraws a loss making contract, then the others will follow by removing their loss making contract. This is the
logic of the Wilson equilibrium concept, where loss making contracts are allowed to
be removed as soon as someone tries to cream skim the market. To make forcing of
withdrawals unattractive, firms offer latent contracts. These latent contracts are used
to cream skim any remaining deviating contract. This feature goes beyond the equilibrium concept of Wilson, in which only first round effects of withdrawals in response
11
to deviating contract offers are considered. However, if the deviating firm anticipates
these responses, further deviating strategies become possible. These strategies are
encountered by latent contracts, whose sole purpose is to cream skim low risks from
the deviator.
Equilibrium with the MWS allocation
We proceed by showing that the MWS allocation can be sustained as equilibrium
allocation. Following the above discussion, latent contracts will be offered alongside
the MWS contracts to prevent deviations.
Definition 1. A contract ω 6= (0, 0) is a latent contract if ω ∈ ∆t for some t ≤ t̂,
but ω ∈
/ arg max uJ (ω).
ω∈∆t̂
Latent contracts are thus contracts - different from the null contract - that are on
offer at some point in the game, but never taken out by a customer. Before coming
to the main result, that an equilibrium exists in which each customer receives his
respective MWS contract, we first show that this indeed requires latent contracts.
Lemma 1. The MWS allocation, when cross-subsidizing, cannot be sustained as
equilibrium allocation without at least one latent contract.
Proof. See Appendix.
Proposition 1. There exists a pure strategy subgame perfect Nash equilibrium where
every individual obtains her respective MWS contract in stage 3.
Proof. See Appendix.
In equilibrium, in stage 1, all but two firms offer the MWS contracts and additionally
the RS contracts as well as a continuum of contracts that lie on the L-type fair
insurance line and give the L-type equal or lower expected utility than her MWS
contract but equal or higher expected utility than her RS contract. We name this
continuum of contracts ‘LR contracts’.13 These contracts are shown in Figure 2. Two
firms offer only the LR contracts.
If all firms make this equilibrium offer in stage 1, there will be no contract withdrawal
in stage 2 and the game ends. If instead there is a deviation in stage 1, in stage 2 in
13
Note that the LR contracts include the low risk RS contract. We say that firms offer the LR
contracts and the RS contracts to ease exposition.
12
w2
uL
uH
uH
H
ωM
WS
L
ωM
WS
H
ωRS
LR contracts
L
ωRS
w02
w01 w1
Figure 2: Contracts on offer in equilibrium
each round t each firm computes the virtual profit it would make if stage 2 ended
after round t − 1, and, if it makes a loss, withdraws the loss-making contract(s), but
does not withdraw any contracts if it makes zero expected profits.
This strategy follows the Wilson logic and supports the MWS allocation for the following reasoning: As discussed in the previous subsection, a simple cream-skimming
deviation which consists in offering a contract like ωA in Figure 3 to attract low
risks only is prevented as other firms would withdraw their MWS contracts since
they become unprofitable. Similarly, as stage 2 ends endogenously, a deviation that
involves the withdrawal of the H-type MWS contract in some round by any firm is
prevented as all other firms would absorb too much H-type demand and become unprofitable, so they withdraw their loss-making H-type MWS contract subsequently as
well. Then the L-type MWS contract attracts both types, becomes unprofitable, and
is withdrawn. Therefore a strategy of only serving L-types and not cross-subsidizing
H-types is prevented.
The more subtle deviation strategy, forcing of withdrawals, is prevented by the LR
contracts: Firms will withdraw those contracts from the set of LR contracts that
would be taken up by H-types and hence be loss-making, but will leave exactly those
LR contracts that would not be taken up by H-types but only by L-types and hence
cream-skim the L-types from any deviating contract like ωB .14 This type of devia14
Note that the LR contracts are a continuum of contracts as we do not restrict the set of feasible
contracts Ω. With a discrete contract grid, the LR contracts would be a finite set, and we could
restrict stage 1 offers and consequently stage 2 contract withdrawals to be finite. Thus, Proposition
1 does not depend on the game being infinite.
13
tion and the pattern of withdrawal according to the equilibrium strategy is shown in
Figure 3. The latent LR contracts will make any deviation with contracts “above”
the RS contracts unattractive.
To prevent deviations “below” the RS contracts, the RS contracts are offered. This
however entails a further strategic complication: The high risk RS contract is, when
taken out by the low risks as well, profit-making.15 Thus, in a subgame where no
contracts above the RS contracts are offered anymore, firms that have offered both
RS contracts have an incentive to withdraw the low risk RS contract in order to make
a higher profit by pooling all risks on the high risk RS contract. To prevent this type
of deviation, there are the two firms offering only the LR contracts, which include
the low risk RS contract. For these two firms it is subgame perfect not to withdraw
the low risk RS contract. With the presence of these two firms offering only the LR
contracts, the strategy profile of the other firms is subgame perfect at the withdrawal
stage.16
w2
MWS contracts will be
withdrawn first
contracts withdrawn
if deviator withdraws
cream-skimming contract
ωA only
ωB
ωA
deviating contract menu
w02
w01 w1
Figure 3: Pattern of contract withdrawal following a deviation
As emphasized before, latent contracts are offered: the RS contracts and the LR
contracts. A standard criticism of latent contracts is that they are loss-making off
the equilibrium path.17 Note that this is not the case here: If, off the equilibrium
path, latent contracts would be the best available contracts on offer for some type,
15
The low risk type might prefer the high risk RS contract over remaining uninsured.
There are two firms, instead of just one firm, offering only the LR contracts, to prevent deviations where any of these two firms offers the potentially profit-making high risk RS contract in
stage 1.
17
The debate on latent contracts is reviewed in Attar, Mariotti, and Salanié (2011).
16
14
they would either not be loss-making, or they would be withdrawn such that they
cannot be chosen in stage 3. In particular, either the LR contracts are taken up only
by low risks, or, potentially in more than one round, they are withdrawn. The other
latent contracts, the RS contracts, are never withdrawn, however, they are individually zero-profit making anyway.
The above Proposition provides an existence result for an equilibrium with an allocation that yields zero expected profits and is second-best efficient.18 It has a particular
property, namely that on the equilibrium path no contract is withdrawn. If one allows for contract withdrawal on the equilibrium path, further outcomes are possible,
in particular there exist equilibria where firms share positive profits.
4
Full Characterization of equilibrium allocations
Any nonnegative profit making, incentive compatible contract menu can be supported as the equilibrium allocation. To prevent stage 1 deviations, i.e. deviating
initial contract menus, some, at least two, firms offer only MWS, RS and LR contracts and withdraw these subsequently. The logic is that given that these firms only
offer contract menus with which they make at most zero profits it is a credible threat
that they do not withdraw these if they observe a stage 1 deviation. To prevent deviations in stage 2 that aim at withdrawing some contracts to make a higher profit,
e.g. pooling all risks on the high risk contract of the intended equilibrium allocation,
there are as for Proposition 1 two firms that offer only the LR contracts, as well as a
contract that would attract only low risks if the low risk contract from the intended
equilibrium allocation were withdrawn.
Thus, we can proceed to show that any allocation that is individually rational, incentive compatible and yields nonnegative profits to firms can be sustained as equilibrium
allocation. Let Φ be the set of such allocations:
Φ :={(ω H , ω L ) ∈ Ω2 |uH (w01 , w02 ) ≤ uH (ω H ), uL (w01 , w02 ) ≤ uL (ω L ), uH (ω H ) ≥ uH (ω L ),
γ[(1 − pH )(w01 − w1H ) + pH (w02 − w2H )]+
(1 − γ)[(1 − pL )(w01 − w1L ) + pL (w02 − w2L )] ≥ 0
and denote an element of Φ by φ.
18
Proposition 1 is shown for the two-type case. In Appendix B, we discuss construction of the
MWS contracts and latent contracts and provide an example for the n-type case.
15
Proposition 2. (i) Any φ ∈ Φ can be sustained as equilibrium allocation in a pure
strategy subgame perfect equilibrium of the game.
(ii) For any equilibrium allocation (ω H∗ , ω L∗ ) it holds that (ω H∗ , ω L∗) ∈ Φ.
Proof. See Appendix.
Although there are different categories of allocations, for instance such that are crosssubsidizing from high to low risks19 , the crucial point in the construction of equilibrium for each of the categories of allocations is that there are (i) some (inactive)
firms that offer menus which are not profit-making and include the MWS contracts
such that it is a credible threat that they do not withdraw MWS contracts if a firm
deviates in stage 1. Furthermore, (ii) there are some (inactive) firms that offer menus
such that in the withdrawal stage, certain (potentially loss-making) contracts from
the intended equilibrium allocation cannot be profitably withdrawn.20
Multiplicity of equilibrium allocations
In our Wilson game, the MWS allocation can be sustained as equilibrium allocation,
as shown in Proposition 1. However, the possibility of individual contract withdrawal
to prevent cream-skimming deviations also allows for other equilibrium allocations
as illustrated in Proposition 2 above.
Allowing for contract withdrawal costs will however not alter the conclusion that
multiple equilibria are possible.21 As argued above, equilibria exist where all firms
make a positive profit in stage 3. Although they have to withdraw contracts on
the way, as long as contract withdrawal costs are small, the profits in stage 3 will be
sufficient to compensate the costs of withdrawal. That is different from the analysis in
Netzer and Scheuer (2014), where positive profits in equilibrium can only be sustained
19
There are four different categories of allocations: First, allocations which give the high risks a
higher expected utility than their MWS contract. Second, allocations which give the high risks an
expected utility equal or lower than their MWS contract and where losses are made on high risks.
Third, allocations which give the high risks an expected utility equal or lower than their MWS
contract but where each contract individually makes nonnegative profits, and fourth, allocations
which give the high risks an expected utility equal or lower than their MWS contract and where
losses are made on low risks.
20
Note that some allocations, those where the H-type contract yields H-types a higher expected
utility than the MWS contracts and those with cross-subsidization from high to low risks, cannot
be sustained with a discrete contract space and finite stage 1 contract offers since they require
the threat of indefinite contract withdrawal. Other allocations, however, including profit-making
allocations, could be sustained in this variant of the game with a finite game structure.
21
Another potential ’solution’ to the problem of multiplicity could be entry. We defer the discussion why entry cannot preclude multiplicity to section 5 where we allow for contract addition.
16
if some firms enter the market and then withdraw completely. In their model with
positive withdrawal costs, this exit strategy is unprofitable. However, as in our case
individual contracts can be withdrawn without exiting the market completely, this
reasoning does not apply here. If contract withdrawal costs are very large, in effect
rendering contract withdrawal infeasible, we are back to the original RS model. In
this model, which is equivalent to our model without stage 2, an equilibrium in pure
strategies does not exist as long as the MWS allocation is different from the RS
allocation.
Equilibria with the MWS allocation are nevertheless special insofar as they are the
only equilibria where no contracts are withdrawn on the equilibrium path. This is
shown in the following proposition.
Proposition 3. Any equilibrium that induces an allocation different from the MWS
allocation requires the removal of at least one contract by a firm on the equilibrium
path.
Proof. See Appendix.
At first glance, this might appear to be a technical peculiarity. However, by
applying the insight of the model to real insurance markets this feature becomes
essential: Real insurance markets do consist of many periods of contract offering,
trading and possibly contract withdrawing. One could interpret the equilibrium with
the MWS allocation as a sustainable outcome, where insurers offer MWS contracts
as long as no deviating contracts are offered. If something happens, e.g. entry or
different contracts are offered, then insurers respond by withdrawing contracts. All
the other equilibria of our model are not amenable to such an interpretation. The
immediate implication of these equilibria would be that firms regularly withdraw
contracts, and transactions do only take place after rounds of withdrawals. It is hard
to reconcile these outcomes with real world markets.
5
Riley’s conjecture
So far, we have considered contract withdrawal in stage 2 in the spirit of Wilson. In
this section, we enlarge the action space and allow for also offering new contracts in
stage 2. This is the dynamic proposed in Riley (1979)’s ‘reactive equilibrium’ concept, in which instead of contract withdrawal, firms anticipate that contracts will be
added in response to a deviation. Applying the reactive equilibrium concept results
17
in the RS allocation. Engers and Fernandez (1987) generalize the reactive equilibrium concept and give a game-theoretic interpretation. In their game, contracts are
repeatedly added sequentially, with the order of moves being exogenously given, and
customers choose once contract addition has ended. Engers and Fernandez (1987)
show that this game has a multiplicity of perfect Nash equilibrium outcomes.22
In this section, we analyze the case where contracts can be both withdrawn and
added before customers choose, thereby embedding a variant of the game in Engers
and Fernandez (1987) into our contract withdrawal game. In a survey, Riley (2001, p.
448) conjectures that in a game where firms are allowed to either add or drop offers
in the second stage “both the Wilson and reactive equilibria are a Nash equilibrium
of this new game”. As Proposition 3 below shows this is true. Consider the game
with the following modification:
Stage 2’: Stage 2 consists of t = 1, 2, ... rounds. In each round t, each firm f ∈ F
can withdraw a set from its remaining contracts and add any set of contracts to
the remaining contracts. After each round, firms observe the contract offers of all
firms. Again, denote by Ωft firm f ’s contract set on offer at the end of t. If, for
any t, Ωft = Ωft−1 for all f ∈ F , this stage ends.
f
A strategy for a firm f is a sequence of maps {ᾱtf }∞
t=0 where each ᾱt maps histories
in sets of contracts, i.e.
ᾱtf : Ht → P(Ω)
ht 7→ Ωft
Ωft compact for all t ≥ 0. Different from the withdrawal game where Ωft ⊆ Ωft−1 for all
t > 0, here there is no restriction regarding the contract offers. Note that by allowing
for contract additions, the game automatically has an infinite horizon, even if the
contract space were discrete. This extension of the action space does not eliminate
equilibrium allocations, in fact, an allocation can be sustained in equilibrium if and
only if it was an equilibrium allocation in the withdrawal game.
22
Note that there are important differences between the contract addition and contract withdrawal games: While firms make losses off the equilibrium path in any equilibrium with crosssubsidization under pure contract addition as in Engers and Fernandez (1987), in our games with
contract withdrawal, firms do not make losses off the equilibrium path.
18
Proposition 4. An allocation is an equilibrium allocation in the extended game that
allows for adding contracts in stage 2 if and only if it is an equilibrium allocation in
the original withdrawal game.
Proof. See Appendix.
It is easy to see that an allocation that cannot be sustained in the withdrawal game,
i.e. loss-making allocations, cannot be sustained in equilibrium. To see why any
allocation in the withdrawal game can be supported as equilibrium allocation in the
extended game, pick an equilibrium in the original game with the corresponding
equilibrium strategy of firms. Then, consider that firms have the same strategy, with
the addition that whenever they observe any new contract offer by any other firm in
stage 2, round t, then in round t + 1 they add the complete set of contracts offered in
stage 1 in the equilibrium strategy. That way, if, e.g. in a profit-making equilibrium,
a firm attempts to make a profit by offering a contract that profitably attracts the
whole population, this strategy replicates, in round t + 1, any possible configuration
of contract offers at the end of stage 1 in the original game. However, then there is
no profitable deviation since it was an equilibrium in the original game. This result
allows us to formally confirm Riley (2001)’s conjecture:
Corollary 1. In the game in which contracts can be withdrawn and added in stage
2, both the MWS and RS allocation can be sustained as equilibrium allocations.
Note that our analysis of contract addition also sheds light on the question whether
entry can address the problem of multiplicity of equilibrium allocations. Entry, in its
general form, is not different from contract addition in the game where firms can offer
and withdraw contracts repeatedly. Thus, in this framework multiplicity of equilibria
still exists with entry. Furthermore, a restricted form of entry in the Wilson game,
where firms after they have entered the market can only withdraw contracts, will not
alter this conclusion. Suppose there are finitely many firms which might enter. Then
the firms could keep all contracts on offer until all firms have entered the market.
Only then they will start to withdraw contracts, so the logic of Proposition 2 still
applies. Suppose that there is potentially infinite entry. But in that case, further
entry can be used a a threat to sustain equilibria. I.e. for any equilibrium allocation,
if a firm enters with deviating contracts then this firm will be penalized by further
entry of firms, which e.g. enter with the MWS contract on offer, making profitable
deviations infeasible.
19
6
Conclusion
In competitive markets with adverse selection and common values, existence of an
equilibrium in which the allocation is second-best efficient is often invoked by referring
to Wilson (1977)’s notion of “anticipatory equilibrium”. However, the anticipatory
equilibrium lacks a sound game-theoretic foundation to date. We propose a game
of contract withdrawal that models the strategic interaction behind the anticipatory equilibrium by augmenting the seminal Rothschild and Stiglitz (1976) model by
an additional stage in which firms can withdraw contracts (repeatedly) after observation of competitors’ contract offers. We show that an equilibrium always exists
where consumers obtain their respective Miyazaki-Wilson-Spence (MWS) contract,
i.e. second-best efficiency can be achieved for any share of high-risk types in the
population. However, contrary to intuition, the game-theoretic analysis of the Wilson concept is not that straightforward: Firstly, latent contracts have to be offered
alongside the MWS contracts for the Wilson logic to work. Furthermore, contract
withdrawal also gives rise to more subtle strategies which allow for allocations in
which firms make positive profits. This remains valid if, besides contract withdrawal,
additional contracts can be offered in the second stage. We can thereby verify that,
as suggested by Riley (2001), both the MWS and RS allocations are equilibria in this
game. Interestingly, cross-subsidization can prevail in equilibrium although firms can
withdraw individual (and, in particular, loss-making) contracts.
20
Appendix
A Proofs
Notations H
H
L
L
Let ΩM W S := ωM
W S , ωM W S denote the set of MWS contracts, ΩRS := ωRS , ωRS
denote the set of RS contracts and
L
L
L
L
ΩLR := ω ∈ Ω|uL (ω) ≤ uL (ωM
W S ), u (ω) ≥ u (ωRS )
and (1 − pL )(w01 − w1 ) + pL (w02 − w2 ) = 0
the (compact) set of contracts that lie on the L-type fair insurance line and yield
an L-type an expected utility that is not lower than expected utility from her RS
contract but not higher than her expected utility from her MWS contract. Note that,
L
of course, ωRS
∈ ΩLR .
Furthermore, let
L
H
H
L
ΩCS := {ω ∈ Ω|uL (ω) > uL (ωM
W S ), u (ω) < u (ωM W S )
and (1 − pL )(w01 − w1 ) + pL (w02 − w2 ) > 0}
denote the cream-skimming contract set with respect to the MWS contracts. We
denote the virtual profit of firm f if stage 2 would end after t − 1 by
H
H
π f (Ωt−1 ) := γ[(1−pH )(w01 − w̄1,t−1
)+pH (w02 − w̄2,t−1
)](1/k̄ H (Ωt−1 ))1I{ω̄H
t−1 ∈
L
L
(1 − γ)[(1 − pL )(w01 − w̄1,t−1
) + pL (w02 − w̄2,t−1
)](1/k̄ L (Ωt−1 ))1I{ω̄L
t−1 ∈
Ωft−1 } +
Ωft−1 }
where 1I is an indicator function. Similarly, we denote by
H
H
π f,H (Ωt−1 ) = γ[(1 − pH )(w01 − w̄1,t−1
) + pH (w02 − w̄2,t−1
)](1/k̄ H (Ωt−1 ))1I{ω̄H
t−1 ∈
Ωft−1 }
and
L
L
π f,L (Ωt−1 ) = (1 − γ)[(1 − pL )(w01 − w̄1,t−1
) + pL (w02 − w̄2,t−1
)](1/k̄ L (Ωt−1 ))1I{ω̄L
t−1 ∈
the virtual profits on H-types and L-types respectively.
\LR
Furthermore, let ∆t := ∆t \ ΩLR denote the set of contracts offered in t excluding
J
J
, ẅ2,t
) the contract such that
the LR contract set. We denote by ω̈tJ = (ẅ1,t
ω̈tJ ∈ arg max uJ (ω)
\LR
ω∈∆t
and
J
J
ẅ2,t
≥ w̃2,t
∀ ω̃tJ ∈ arg max uJ (ω)
\LR
ω∈∆t
ω̈tJ is the highest coverage contract maximizing J-type utility among the contracts offered in t when the LR contract set is removed. Note that “above” MWS and “below”
f
H
H
H
RS contracts, ω̈tJ and ω̄tJ coincide. Then, let Ω̂f,LR
t−1 := {ω ∈ Ωt−1 |u (ω) ≤ u (ω̈t−1 )}.
21
Ωft−1 }
f
Ω̂f,LR
t−1 is the set where all contracts are removed from Ωt−1 which are more attractive
for the H-types than the best contract, outside the set ΩLR , currently on offer.
Lastly, to incorporate no insurance in a simple manner, we impose that (w01 , w02 ) ∈
Ωft for all f ∈ F and t ≥ 0. To ease exposition, we will not explicitly add (w01 , w02 )
as part of stage 1 and stage 2 contract menus in the following proofs, but just assume
that no insurance is ‘offered’ by each firm in stage 1, and in every round in stage 2,
alongside the stated contract menus.
Proof of Lemma 1.
Assume that an equilibrium with the MWS allocation, i.e. an equilibrium in which
each customer of type J receives her respective MWS contract in stage 3, exists without latent contracts. Then, for all f ∈ F and t, Ωft ⊆ ΩM W S , and every f ∈ F
makes at most zero profits. We will show that a profitable deviation exists. Assume
firm f¯ deviates and offers a contract menu {ωcs , ωp } in stage 1 where ωcs ∈ ΩCS and
ωp makes a strictly positive profit if taken out by the entire population but is not
preferred by high risks to their MWS contract. Since the contract sets of each firm
are finite, stage 2 ends for sure after some t̂ < ∞. Then if ΩM W S ∩ ∆t̂ 6= ∅ , then
the deviator makes a strictly positive profit with low risks by keeping ωcs on offer for
all t ≤ t̂. If ΩM W S ∩ ∆t̂ = ∅ , then there exists t̃ such that ΩM W S ∩ ∆t̃−1 6= ∅ and
¯
ΩM W S ∩ ∆t̃ = ∅. Then, f¯ makes a strictly positive profit by choosing Ωf = {ωcs , ωp }
for all t ≤ t̃ and withdrawing ωcs in t̃ + 1 while keeping ωp on offer.
Proof of Proposition 1.
Equilibrium strategy
The equilibrium strategy of firms is the following: In stage 1, firms f ∈ F \ {n − 1, n}
offer Ωf0 = ΩM W S ∪ ΩRS ∪ ΩLR . Firms n − 1 and n offer the LR contracts, i.e.
Ω0n−1 = Ωn0 = ΩLR . In stage 2, we fully describe the strategy at all subgames when in
stage 1 at most one firm did offer a contract set different from those specified above.
If at most one firm f¯ 6= f deviated in stage 1, the strategy of firm f ∈ F in round
t > 0 specifies
 f
if π f (Ωt−1 ) ≥ 0;
 Ωt−1
f
f
αt (ht ) =
Ωt−1 \ ΩM W S if π f (Ωt−1 ) < 0 and Ωft−1 ∩ ΩM W S 6= ∅;
 f,LR
Ω̂t−1
if π f (Ωt−1 ) < 0 and Ωft−1 ∩ ΩM W S = ∅;
Note that the stage 2 strategy is the same for all firms, i.e. including firms n − 1
and n who only offer the LR contracts in stage 1. The strategy specifies that if firms
make a nonnegative virtual profit after round t − 1, e.g. on MWS contracts or on RS
contracts, they do not withdraw any contracts in round t. If a firm makes a virtual
loss after round t − 1, and the firm still offers a MWS contract, the firm withdraws
the MWS contract menu. If a firm makes virtual losses and does not offer any MWS
contract, then the firm withdraws contracts such that the remaining (compact) contract set does not contain any contract that is preferred by high risks to the best
high risk contract when LR contracts are assumed not on offer, but leaves exactly
the contract that cream-skims low risks with respect to this contract. Observe that
22
the strategy specifies that if there is one firm offering a contract set in stage 1 that is
different from ΩM W S ∪ΩRS ∪ΩLR , both RS contracts always remain on offer. Observe
also that a firm always withdraws a nonempty set of contracts if it makes a virtual loss.
No profitable deviation
If all firms follow the above strategy, no firm withdraws any contract in t = 1 and
stage 2 ends after t = 1, firms make zero expected profit and a customer of type J
receives her J-type MWS contract.
We will first show that there is no profitable deviation. We will then show that the
strategy profile/equilibrium is subgame perfect (at the withdrawal stage). To show
that there is no profitable deviation we proceed in two steps: First, we show that
a deviator serves some H-types. Second, we show that if the deviator serves some
H-types, she cannot be making a strictly positive profit.
¯
¯
¯
Consider a deviating firm f¯ that offers Ωf0 in stage 1 and has a strategy α̂f : ht 7−→ Ωft
in stage 2. Let Ωft̂ be the final set of contract offers of firm f , i.e. Ωft̂ = Ωft̂−1 ∀ f ∈ F .
as otherwise, according to the
Then it must be that π f (Ωt̂−1 ) ≥ 0 ∀ f ∈ F \ f¯ equilibrium strategy defined above, a firm f ∈ F \ f¯ would withdraw a nonempty
set of contracts in t̂ and t̂ would not be the final round in stage 2.
¯
¯
Now assume π f (Ωt̂ ) > 0. As π f (Ωt̂−1 ) ≥ 0 ∀ f ∈ F \ f¯ , we will show that w̄t̂H ∈ Ωft̂ :
¯
Since π f (Ωt̂ ) > 0, f¯ serves some customers. To show that it cannot be possible that
f¯ serves only L-types, assume on the contrary that f¯ only serves L-types. If Ltypes prefer an insurance contract to remaining uninsured, then H-types prefer
to be
insured as well. As f¯ only serves L-types, then at least one firm f ∈ F \ f¯ serves
H-types and the share of L-types among customers at f is less than 1 − γ. There are
three possible cases:
H
¯ that serves H-types with ω H
Case 1 : ω̄t̂H = ωM
W S . Now any firm f ∈ F \ f
MW S
and has a share of L-types among customers that is less than 1−γ does not make a
nonnegative profit.23 This contradicts π f (Ωt̂−1 ) ≥ 0 ∀ f ∈ F \ f¯ .
Case 2 : ω̄t̂H ∈ ΩLR . Any contract ω ∈ ΩLR if taken up by some H-types is lossmaking, independent of whether
it is also taken up by some L-types. This contradicts
πf (Ωt̂−1 ) ≥ 0 ∀ f ∈ F \ f¯ .
H
. Then, from the strategy of firms n − 1 and n, at least one firm
Case 3 : ω̄t̂H = ωRS
H
still has the L-type RS contract on offer when ω̄t̂H = ωRS
. Then, there is no contract
(menu) that f¯ can offer attracting L-types only and making a positive profit, which
is a contradiction.
¯
¯
Hence, ω̄t̂H ∈ Ωft̂ . We will now show that if ω̄t̂H ∈ Ωft̂ , f¯ cannot be making a positive
profit. First, note that the RS contracts remain always on offer. This is because firms
n − 1 and n do not withdraw the low risk type RS contract as long as the high risk
type RS contract is on offer, and all other firms do not withdraw the high risk type
RS contract as long as the low risk type RS contract is on offer. Then, it follows that
H
). There are again three possible cases:
uH (ω̄t̂H ) ≥ uH (ωRS
H
H
H
L
Case 1 : u (ω̄t̂ ) ≥ uH (ωM
been withdrawn by any
W S ). Then, ωM W S will not have
L
¯
¯
firm f ∈ F \ f . As ωM W S is on offer from firms f ∈ F \ f , by construction of the
L
This is because firm f ∈ F \ f¯ at best serves some L-types with ωMW
S , however, since the
share of L-types is less than 1 − γ, this is loss-making.
23
23
¯
MWS contracts, π f (Ωt̂ ) ≤ 0 for the cases that f¯ only serves H-types or both types.
H
L
Case 2 : uH (ω̄t̂H ) < uH (ωM
W S ) and ω̄t̂ ∈ ΩCS , i.e. the preferred L-type contract is in
the cream-skimming contract set with respect to MWS
contracts. Hence, both MWS
¯
contracts are not on offer at any firm f ∈ F \ f as they have been withdrawn,
since firms make losses onMWS
contracts if a cream-skimming contract is on offer.
¯
Thus, any firm f ∈ F \ f does not serve neither L-types nor H-types since it
does not offer any contract w ∈ ΩCS . This implies that with ω̄t̂H the deviator firm
f¯ serves all H-types. However, by construction of the MWS contracts, there is no
incentive compatible menu of contracts with w̄t̂L ∈ ΩCS that is profit-making, hence
πf¯(Ωt̂ ) < 0.
H
L
Case 3 : uH (ω̄t̂H ) < uH (ωM
/ ΩCS . First consider that ω̄t̂H ∈ ΩLR ∪ ΩRLR
W S ) and ω̄t̂ ∈
with
L
H
H
H
ΩRLR :={ω ∈ Ω|uL (ω) ≤ uL (ωM
W S ) and u (ω) ≥ u (ωRS );
(1 − pL )(w01 − w1 ) + pL (w02 − w2 )] < 0},
i.e. ω̄t̂H is such that it is making zero or negative profits even when taken out by L¯
/ Ω ∪ ΩRLR .
tyes. Then it immediately follows π f (Ωt̂ ) < 0. Therefore consider ω̄t̂H ∈
LR
L
f
¯
¯
From the strategy of any firm f ∈ F \ f , ω̄t̂ ∈ Ωt̂ ∀ f ∈ F \ f as in every t
firms f ∈ F \ f¯ leave the largest (compact) subset from ΩLR , including the L-type
RS contract, on offer that attracts low risks but does not give high risks a higher
expected utility than that of the most preferred contract of H-types from ∆t \ ΩLR .
Then, either the deviator’s contract for the low risks is loss making, or the share of
low risk customers at f¯ is lower than 1 − γ while f¯ serves all high risks such that
¯
π f (Ωt̂ ) ≤ 0.
¯
Hence, π f (Ωt̂ ) ≤ 0 which is a contradiction.
Subgame perfection
When in stage 1 not more than one firm offered a contract set different from those
specified above for firms n−1 and n and all other firms respectively, then the strategy
prescribes that in t, firms do not withdraw any contract if they make a nonnegative
virtual profit, but always withdraw the virtually loss-making contracts in t if they
make a virtual loss. Following a stage 1 contract menu ΩM W S ∪ ΩRS ∪ ΩLR for firms
f ∈ F \ {n − 1, n} and ΩLR for firms n − 1 and n, this strategy profile is subgame
perfect for the following reason: First consider firms f ∈ F \ {n − 1, n}. If a firm
were to makes a virtual loss, it withdraws the loss-making contract(s). Except for the
MWS and RS high risk type contract, none of the offered contracts is profit-making
if taken out by the whole population. Now the strategy profile is such that a firm
f ∈ F \ {n − 1, n} can never make a positive profit on either of the two contracts:
As long as the MWS high risk contract is on offer either the low risk MWS or the
cream-skimming LR contracts are still on offer as well by at least one firm, n − 1 or
n, thus low risks will not buy the high risk MWS contract. The high risk RS contract will also never be bought by the whole population, as the low risk RS contract
remains always on offer by at least firm, n − 1 or n. There are two firms offering
only the LR contracts to ensure that the strategies of all firms are subgame-perfect
even if there is a stage 1 deviation, i.e. not offering only the LR contracts, by either
firm n − 1 and n. Having two firms ensures that there is at least one firm offering
24
only the LR contracts such that withdrawal of contract by firms f ∈ F \ {n − 1, n} is
subgame-perfect. For firms n − 1 and n, since they only offer the LR contracts and
can thus not make a strictly positive profit on any contract, it is subgame-perfect to
leave non-loss-making LR contracts, including the low risk RS contract, on offer.
We have specified that in stage 2, at all subgames reached when in stage 1 more than
one firm did offer a contract set different from ΩM W S ∪ ΩRS ∪ ΩLR , firms play any
subgame perfect equilibrium strategy of the subgame starting in stage 2. Thus, the
strategy profile is subgame perfect for these subgames by definition.
Proof of Proposition 2.
(i) We construct the corresponding equilibria for any allocation φ = (ωφH , ωφL ) ∈ Φ.
H
L
Case I: φ such that uH (ωφH ) ≤ uH (ωM
W S ) and ωφ makes nonnegative profits with low
risks.
First, we consider the case that φ is such that the high risk contract does not give
high risks a higher expected utility than the high risk MWS contract and there is
no cross-subsidization from high to low risks. Among these, there are allocations
that give high risks a higher expected utility than their RS contract such that the
allocation is cross-subsidizing from low to high risks, and some where the high risks
receive an expected utility that is equal or lower than that from their RS contract.
We will construct equilibria in the following way: To prevent stage 1 deviations, some
firms, which we will henceforth call inactive MWS firms, offer exactly the MWS, RS
and LR contracts. For these it will be credible not to withdraw MWS contracts if
they observe a stage 1 deviation, but they will withdraw their offered contract sets
along the equilibrium path. To prevent stage 2 deviations, e.g. by inactive MWS
firms that try to pool all risks on the high risk RS contract, or by a firm offering the
intended equilibrium allocation but withdrawing a contract from this menu in order
to make a higher profit, there will be two further firms, which we will henceforth
call again LR firms. The LR firms offer the LR contracts, as in Proof of Proposition
of 1, and in addition a contract that is zero profit-making when taken out by low
risks, gives low risks a lower expected utility than the low risk contract from φ but
a higher expected utility than the high risk contract from φ, but gives high risks a
lower expected utility than the high risk contract from φ. I.e., when only the intended equilibrium allocation is on offer and the low risk contract from the intended
equilibrium allocation would be withdrawn, this contract would be taken out by low
risks and only low risks, and would not make losses. Last, we let exactly one firm
offer the intended equilibrium allocation. This is to prevent, in a simple manner,
the problem of withdrawal of a loss-making high risk contract by some firm when the
intended equilibrium allocation is cross-subsidizing and offered by more than one firm.
Equilibrium strategies
We specify the strategy in stage 1, and from stage 2 onwards if in stage 1 at most
one player offers a contract set different from equilibrium contracts.24 Let ΩIA :=
ΩM W S ∪ ΩRS ∪ ΩLR . Furthermore, let ω L̂φ denote the contract that solves:
24
These equilibrium contracts might be withdrawn along the equilibrium path.
25
max uL (ω)
ω s.t.
(6)
uH (ωφH ) ≥ uH (ω)
(7)
uL (ωφL ) ≥ uL (ω)
(8)
L
L
(1 − p )(w01 − w1 ) + p (w02 − w2 ) = 0
(9)
The equilibrium strategies of firms are the following: In stage 1, firm 1 sets Ω10 =
{ωφH , ωφL }, firms f ∈ F \ {1, n − 1, n} offer Ωf0 = ΩIA and firms n − 1 and n set
Ω0n−1 = Ωn0 = ΩLR ∪ {ω L̂φ }.25 We will henceforth call all firms f ∈ F \ {1, n − 1, n}
inactive MWS firms. We will call firms n − 1 and n LR firms. The inactive MWS
firms withdraw their contract menus on the equilibrium path, the LR firms withdraw
some but not all contracts on the equilibrium path.26
To ease notation, we will denote histories at the beginning of stage 2 in the following
way: We say ND1 holds if every firm offered the contract menus in stage 1 as specified
by the equilibrium strategy above, i.e. Ω10 = {ωφH , ωφL }, Ωf0 = ΩIA for all firms
f ∈ F \ {1, n − 1, n} and Ω0n−1 = Ωn0 = ΩLR ∪ {ω L̂φ } for firms n − 1 and n. We say D1
holds if some firm did offer a contract menu in stage 1 different from the contract menu
specified by the equilibrium strategy above, i.e. either Ω10 6= {ωφH , ωφL }, or Ωf0 6= ΩIA
for some firm f ∈ F \ {1, n − 1, n} or Ω0n−1 6= ΩLR ∪ {ω L̂Φ } or Ωn0 6= ΩLR ∪ {ω L̂φ }.
If at most one firm did offer a contract set different from those specified above in
stage 1, the strategy of firms f ∈ F \ {1, n − 1, n} in round t > 0 in stage 2 specifies
 f
if D1 holds and π f (Ωt−1 ) ≥ 0;
Ω


 ft−1
Ωt−1 \ ΩM W S if D1 holds, π f (Ωt−1 ) < 0 and Ωft−1 ∩ ΩM W S 6= ∅;
αtf (ht ) =

if D1 holds, π f (Ωt−1 ) < 0 and Ωft−1 ∩ ΩM W S = ∅;
Ω̂f,LR

 t−1
{(w01 , w02 )} if ND1 holds.
The first three lines describe the strategy if in the first stage someone deviated by
offering a different set of contracts; the last contract withdrawal along the equilibrium
path.
If at most one firm did offer a contract set different from those specified above in
stage 1, the strategy of firms n − 1 and n in round t > 0 in stage 2 specifies

f

if π f (Ωt−1 ) ≥ 0;
 Ωt−1
Ω̂f,LR
if π f (Ωt−1 ) < 0 and ωφH ∈ ∆t−1 ;
αtf (ht ) =
t−1

 Ω̂f,LR \ {ω L̂φ } if π f (Ω ) < 0 and ω H ∈
t−1
t−1
φ / ∆t−1 .
Note that the inactive LR firms always leave the contract ω L̂φ on offer as long as
ωφH is on offer. Note further that as long as the high risk RS contract is on offer,
the inactive LR firms also keep the low risk RS contract on offer (from their LR
contracts).
25
26
L̂
Note that ωΦ
may well be a contract from ΩLR .
L̂
Φ
ω will not be withdrawn on the equilibrium path.
26
Lastly, the strategy of firm 1 in round t > 0 in stage 2 specifies
1
Ωt−1
if π 1 (Ωt−1 ) ≥ 0;
1
αt (ht ) =
Ω1t−1 \ {ω φ } if π 1 (Ωt−1 ) < 0;
where ω φ denotes the contract from Ωft−1 on which the virtual losses would be made.
In stage 2, at all subgames reached when in stage 1 more than one firm did offer
a contract set different from those specified above, firms play any subgame perfect
equilibrium strategy of the subgame starting in stage 2.
The above strategies prescribe the following: Firm 1 offers the intended equilibrium
allocation as contract menu. Inactive MWS firms offer the MWS, RS and LR contracts. LR firms offer the LR contracts and ω L̂φ . If a firm deviates in stage 1 by
offering a different contract menu than specified, then inactive MWS firms behave as
in the proof of Proposition 1.
If every firm offers contract menus in stage 1 according to the equilibrium strategy,
the inactive MWS firms withdraw their contract menus in the first round of stage
2.27 Thus, the logic is that inactive MWS firms prevent stage 1 deviations. Contract
menus of LR firms are constructed to prevent stage 2 deviations. First, LR contracts
are left on offer by inactive LR firms such that inactive MWS firms cannot profitably
deviate by not withdrawing the high risk MWS or RS contract. Second, ω L̂Φ is left
on offer as long as the high risk contract from the intended equilibrium allocation is
on offer, such that firm 1 cannot profitably deviate by withdrawing the low risk contract from the intended equilibrium allocation. Firm 1, which in stage 1 only offered
as contract set the equilibrium allocation, leaves the allocation on offer if it makes
nonnegative virtual profits, and withdraws the respective loss-making contract(s) if
they make virtual losses. This latter part is to ensure subgame-perfection.
If all firms follow the above respective strategies, the inactive MWS firms offering the
MWS, RS and LR contracts withdraw their contract menus in t = 1, LR firms withdraw all contracts from the LR contracts that give high risks a higher utility than ωφH
in t = 2, firm 1 does not withdraw any contract and stage 2 ends after t = 3. Firm
1 makes nonnegative profits and a customer of type J receives her J-type contract
from φ.28
No profitable deviation
If there is a stage 1 deviation, inactive firms act as in proof of Proposition 1. The
proof proceeds along the same lines as the proof of Proposition 1 and is therefore
omitted. Thus, it remains to show that there is no profitable deviation in stage 2 if
all firms offer the contract menus in stage 1 as specified in the equilibrium strategy.
First, consider a stage 2 deviation by some firm f¯ ∈ F \ {1, n − 1, n}, i.e. an inactive
MWS firm. In stage 2, f¯ can only profitably deviate if it either pools all risks on
27
Except for the ‘no insurance’ contract, which we impose to always be on offer alongside to
incorporate no insurance in a simple manner.
28
When ωφL is zero profit-making with low risks, ωφL and ω L̂φ coincide such that LR firms serve
some low risks in stage 3. When ωφL is profit-making, LR firms are inactive in the sense that they
only offer latent contracts and do not serve a customer in stage 3.
27
the high risk MWS contract or on the high risk RS contract. By the equilibrium
strategy of firms the inactive LR firms n − 1 and n, pooling all risks on either the
high risk MWS or RS contract cannot occur, since the corresponding cream-skimming
contracts attracting low risks will not be withdrawn by firms n − 1 and n as long as
the high risk MWS or respectively low risk RS contract are still on offer and firms
n − 1 and n thereby do not make losses on these low risk contracts. Next, consider
a stage 2 deviation by firm 1. Firm 1 cannot profitably deviate by not offering a
cross-subsidized high risk contract if φ is cross-subsidizing high risks: Since it is the
only firm offering {ωφH , ωφL }, withdrawing the cross-subsidized contract and pooling
all risks on the low risk contract always yields equal or lower profits than keeping
all contracts from {ωφH , ωφL } on offer. Second, firm 1 cannot profitably deviate by
withdrawing ωφL and keeping ωφH on offer, i.e. poling all risks on ωφH . This is because
as long as ωφH is on offer, the inactive LR firms keep ω L̂φ on offer which attracts
low risks if ωφL is withdrawn. Lastly, no contract offered by the inactive LR firms is
potentially profit-making such that there is no profitable stage 2 deviation for these
firms.
Subgame perfection
Note that inactive MWS firms are indifferent between being inactive or making zero
profits on MWS contracts. Since n ≥ 5, i.e. there are at least two firms offering only
MWS, RS and LR contracts, and there are two inactive LR firms offering only the
LR contracts and ω L̂φ , it is indeed subgame perfect to keep MWS contracts on offer
if a stage 1 deviation, by either firm 1, another inactive firm or an inactive LR firm
is observed. For this type of deviation (stage 1 deviation), withdrawal prescribed by
the equilibrium strategy is subgame-perfect as shown in the proof of Proposition 1.29
Now consider stage 2. First we check subgame-perfection of the strategies of inactive
MWS firms. The strategy prescribes complete withdrawal if there was no stage 1
deviation. Since, with inactive LR firms offering the LR contracts as long as they are
not making a virtual loss with those, there is no contract on which positive profits
can be made by inactive MWS firms with another withdrawal strategy in the continuation game. Then, it is subgame-perfect to withdraw the complete contract menu
in t = 1 if there is no stage 1 deviation. Next, we check the strategy of firm 1 in
the withdrawal phase. Since firm 1 is the only firm offering {ωφH , ωφL }, it is subgameperfect not to withdraw ωφH as long as ωφL is on offer. This is because the profit of a
firm offering any incentive compatible menu would be lower if it were to withdraw the
high risk contract and pool all risks on the low risk contract. Second, it is subgameperfect not to withdraw ωφL since the inactive LR firms offer ω L̂φ as long as ωφH is on
offer and would attract the low risks such that firm 1 cannot pool all risks on ωφH .
Thus, it is subgame-perfect to leave all contracts from {ωφH , ωφL } on offer. Last, we
check strategies in the withdrawal phase of inactive LR firms. They cannot make a
positive profit on any of their offered contracts and withdraw loss-making contracts
29
Note that one part of this is that since there are two inactive LR firms, even with one inactive
LR firm deviating in stage 1 by e.g. not offering the LR contracts, it is subgame-perfect to leave
MWS contracts on offer for inactive firms and not sequentially withdraw some contracts in order to
pool all risks on the high risk MWS or RS contract, since there remains one inactive LR firm only
offering the LR contracts that would cream-skims low risks from any such intended pooling.
28
from LR. Furthermore, they withdraw ω L̂φ if ωφH is not on offer and ω L̂φ is potentially
loss-making. Thus, the strategy is subgame-perfect in the withdrawal stage.
There are two special cases, where the equilibrium strategies as specified above
do not work. One is where the high risks obtain a utility larger than the utility at the
MWS contract. One example for such a scenario would be pooling at full insurance
at the average fair risk premium. The other is where the allocation is such that the
high risk contract makes a profit, while the low risk contract makes a loss.
H
Case II: φ such that uH (ωφH ) > uH (ωM
W S ).
Consider the case that φ is such that the high risk contract gives high risks a higher
expected utility than the high risk MWS contract.30 In that case, the inactive MWS
firms might make positive (virtual) profits with their low risk type MWS contract.
In order to make it optimal for the inactive firms to withdraw the low risk type MWS
contract, the contract set of firm 1 needs to be amended. If it contains a continuum of
contracts, firm 1 can make the ”threat” to withdraw contracts indefinitely if the other
firms do not withdraw their low risk type MWS contract.31 Thus, let ΩLRS denote
the contracts on the low risk fair insurance line between the low risk RS contract
and the no insurance point, i.e. the line segment between these two points32 , and let
ΩfLRSt := {ω ∈ ΩLRS |ω ∈ Ωft }, i.e. the set of contracts from the line segment ΩLRS
that firm f has on offer in t.
Furthermore, in order to slightly adjust stage 1 offers of inactive LR firms and keep
their stage 2 strategy simple, let ΩLR := {ω ∈ ΩLR |uL (ω) ≤ uL (ωφL )}. Note that
ω L̂φ ∈ ΩLR . We specify the strategy in stage 1, and from stage 2 onwards if in
stage 1 at most one player offers a contract set different from equilibrium contracts.
The equilibrium strategies of firms are the following: In stage 1, firm 1 sets Ω10 =
{ωφH , ωφL } ∪ ΩLRS , firms f ∈ F \ {1, n − 1, n} offer Ωf0 = ΩIA and firms n − 1 and n set
Ω0n−1 = Ωn0 = ΩLR . If at most one firm did offer a contract set different from those
specified above in stage 1, the strategy of firms f ∈ F \ {1} in round t > 0 in stage
2 are the same as specified under Case I.
As discussed above, there is a difference in the stage 2 strategy of firm 1, the strategy
of firm 1 in round t > 0 in stage 2 specifies
1
Ωt−1
if π 1 (Ωt−1 ) ≥ 0;
1
αt (ht ) =
1
Ωt−1 \ Ω̂LRSt−1 if π 1 (Ωt−1 ) < 0;
where Ω̂LRSt−1 denotes the open line segment from the (closed) line segment Ω1LRSt−1
whose length is 1/2 of the length of Ω1LRSt−1 and whose endpoints have the lowest
distance to the low risk RS contract.
In stage 2, at all subgames reached when in stage 1 more than one firm did offer
30
These are all allocations that lie in the area between the MWS contracts and the full insurance
fair pooling contract.
31
The allocations are thus allocations that cannot be sustained if the contract grid is discrete
and firms’ contract offers are required to be finite.
32
The choice of this contract set is arbitrary. Note that each of these contracts is loss-making
when taken out by the whole population.
29
a contract set different from those specified above, firms play any subgame perfect
equilibrium strategy of the subgame starting in stage 2.
The above strategies prescribe the following: Firm 1 offers the intended equilibrium
allocation as contract menu and additionally the continuum of contracts on the line
segment between the low risk RS contract and the no insurance contract. As long as
firm 1 makes virtual losses with its contract menu, it withdraws some contracts on
the line segment. Thus the game would go on indefinitely if firm 1 does not make
nonnegative virtual profits with its contract menu. Note that firm 1 never withdraws
{ωφH , ωφL }. The inactive MWS firms offer the RS, MWS and LR contracts as above
and inactive LR firms offer the LR contracts that do now give low risks an expected
utility higher than that from ωφL . If there is no stage 1 deviation, then inactive
MWS firms withdraw their contract menus. If a stage 1 deviation is observed, then
inactive MWS firms have the same strategy as firms in the proof of Proposition 1.
Furthermore, if there is a stage 2 deviation where inactive firms do not withdraw the
low risk MWS contract such that firm 1 would make a loss if stage 2 ended, in each t
firm 1 withdraws a (open) nonempty set of contracts from the line segment Ω1LRSt−1
such that stage 2 does not end as long as the low risk MWS contract is still on offer
from an inactive firm.
If all firms follow the above respective strategies, inactive MWS firms withdraw their
contract menus in t = 1, firm 1 withdraws an open line segment from the LRS contracts in t=1, inactive LR firms do not withdraw any contract in t = 1, there is no
contract withdrawal in t = 2 and stage 2 ends after t = 2. All firms except firm 1
do not sell any contracts, firm 1 makes nonnegative profits and a customer of type J
receives her J-type contract from φ.
No profitable deviation
If there is a stage 1 deviation, inactive MWS firms play as in Proof of Proposition 1.
Firm 1 withdraws a nonempty set of contracts in every t if it makes virtual losses at
the end of t − 1. Thus, either there is a stage 1 deviation by firm 1, and the proof
proceeds along the same lines as proof of Proposition 1 and is therefore omitted, or
there is a stage 1 deviation by any other firm. Then, however, either firm 1 makes
a virtual loss and stage 2 does not end, or firm 1 does not make a loss. Then, since
firm 1 leaves {ωφH , ωφL } always on offer and does not make a loss, there is no contract
menu that, if offered alongside {ωφH , ωφL }, makes a positive profit such that there is
no profitable deviation.
It remains to show that there is no profitable stage 2 deviation. First, consider a
stage 2 deviation by an inactive MWS firm. Since firm 1 withdraws a nonempty set
of contracts in every t when the low risk MWS contract was still on offer at the end of
t−1, stage 2 does not end, and firms receive zero profits if the low risk MWS contract
remains on offer. Thus, no inactive firm can profitably deviate by leaving the low
risk MWS contract on offer. Furthermore, an inactive MWS firm cannot deviate by
pooling all risks on the high risk MWS or RS contract. This is since either firm 1
makes a virtual loss (and stage 2 does not end), or firm 1 does not make a virtual
loss, but then 1 serves all high and low risks, which contradicts an inactive MWS
firm pooling risks on the high risk MWS or RS contract. Next, consider a stage 2
deviation by firm 1. As in Case I, firm 1 cannot profitably deviate by withdrawing
30
ωφL , since then low risks would be attracted by ωφL̂ offered by the inactive LR firms.
Inactive LR firms cannot profitably deviate in stage 2 with the same reasoning as in
Case I.
Subgame perfection
Note that all firms except firm 1 are indifferent between being inactive, making zero
profits on MWS contracts and making zero profits because stage 2 does not end.
We will only go through subgame-perfection for the subgames that differ from those
in Proposition 1 and Case I above. For inactive MWS firms, it is subgame-perfect
to withdraw the low risk MWS contract and with it the complete contract menu,
as otherwise they make zero profits since stage 2 does not end. For firm 1, it is
subgame-perfect not to let stage 2 end as long as it makes a negative profit.
Case III: φ such that ωφL makes losses with low risks.
With a loss-making low risk contract, a problem is that any firm offering {ωφH , ωφL }
would like to withdraw this loss-making low risk contract, either to pool all risks on
the high risk contract or not serve the low risks, both of which would yield higher
profits.33 Whereas pooling the low risks on the profitable high risk contract can
be prevented in a similar way as in Case I above, not serving the low risks with
the loss-making contract cannot be prevented by latent contracts, as the low risk
contract is already loss-making. However, the withdrawal of the low risk contract
can be prevented by amending the contract set of some other firm, e.g. an inactive
LR firm. If it contains a continuum of contracts, this firm can make the ”threat” to
withdraw contracts indefinitely if firm 1 withdraws ωφL .34 Thus, let ΩN I denote the
contracts on the low risk fair (negative) insurance line between, and including, the
L
no insurance point (w01 , w02 ) and the contract (w01 + 1, w02 − 1−p
), i.e. the closed
L
p
line segment between these two points.35 Let ΩfN It := {ω ∈ ΩN I |ω ∈ Ωft }, i.e. the set
of contracts from the line segment ΩN I that firm f has on offer in t.
We specify the strategy in stage 1, and from stage 2 onwards if in stage 1 at most
one player offers a contract set different from equilibrium contracts. The equilibrium
strategies of firms are the following: In stage 1, firm 1 sets Ω10 = {ωφH , ωφL }, firms
f ∈ F \ {1, n − 1, n} offer Ωf0 = ΩIA , firm n − 1 sets Ω0n−1 = ΩLR and firm n sets
Ωn0 = ΩLR ∪ ΩN I . If at most one firm did offer a contract set different from those
specified above in stage 1, the strategy of firms f ∈ F \ {n} in round t > 0 in stage
2 are the same as specified under Case I.
33
The difference to a menu with a loss-making high risk contract is that a firm, being the only
firm offering the menu, would not withdraw the loss-making high risk contract as then high risks
would take the low risk contract, which yields lower profits.
34
The allocations are thus allocations that cannot be sustained if the contract grid is discrete
and firms’ contract offers are required to be finite.
35
The choice of this contract set is arbitrary. Note that each of these contracts yields negative
insurance and would never be taken out by any consumer.
31
The strategy of firm n in round t > 0 in stage 2 specifies
 n
if ωφL ∈ ∆t−1 and π n (Ωt−1 ) ≥ 0 ;
 Ωt−1
Ω̂n,LR
if ωφL ∈ ∆t−1 and π n (Ωt−1 ) < 0;
αtn (ht ) =
 t−1
Ωnt−1 \ Ω̂N It−1 if ωφL ∈
/ ∆t−1 ;
where Ω̂N It−1 denotes the open line segment from the (closed) line segment ΩnN It−1
whose length is 1/2 of the length of ΩnLRSt−1 and whose endpoints have the lowest
distance to the no insurance point.
In stage 2, at all subgames reached when in stage 1 more than one firm did offer
a contract set different from those specified above, firms play any subgame perfect
equilibrium strategy of the subgame starting in stage 2.
The above strategy prescribes the following: Firm n offers the LR contracts and additionally a continuum of contracts on the low risk fair insurance line with negative
insurance. If ωφL ∈
/ ∆t−1 , firm n withdraws a nonempty set of contracts in every t,
L
i.e. if ωφ is not offered in stage 1 or withdrawn in some t, stage 2 does not end and
all firms make zero profits.
If there is no stage 1 deviation, then inactive MWS firms withdraw their contract
menus. If a stage 1 deviation is observed, then inactive MWS firms have the same
strategy as firms in the proof of Proposition 1. If there is a stage 2 deviation by inactive MWS firms, firms n − 1 and n don’t withdraw some cream-skimming contracts.
If there is a stage 2 deviation by firm 1 in the form of withdrawal of ωφL , stage 2 does
not end.
If all firms follow the above respective strategies, inactive MWS firms withdraw their
contract menus in t = 1, firms n − 1 and n withdraw their LR contracts in t = 2,
firm 1 does not withdraw any contract and stage 2 end after t = 3. All firms except
firm 1 do not sell any contracts, firm 1 makes nonnegative profits and a customer of
type J receives her J-type contract from φ.
No profitable deviation
If there is a stage 1 deviation, inactive MWS firms play as in Proof of Proposition
1. The proof proceeds along the same lines as proof of Proposition 1 and is therefore
omitted.
It remains to show that there is no profitable stage 2 deviation. There is no profitable
stage 2 deviation for an inactive MWS firm or for the inactive LR firms n − 1 and
n following the same reasoning as in Case I. Last, consider a stage 2 deviation by
firm 1. The only potentially profitable deviation is withdrawal of ωφL . Then, however,
from the strategy of firm n, stage 2 does not end, and firm 1 receives zero profits such
that the deviation is not profitable.
Subgame perfection
We only go through the part that differs from Case I. Note that firm n can never
make a positive profit and is indifferent between being inactive, making zero profits
on low risks with some LR contract and making zero profits because stage 2 does not
end. Then, it is subgame-perfect for firm n not to let stage 2 end if ωφL is withdrawn.
32
(ii) We show that any allocation that is not an element of Φ cannot be an equilibrium
allocation. An allocation that is not an element of Φ is not individually rational for
consumers, incentive compatible or loss-making. If the allocation is not individually
rational or incentive compatible, some type does not buy its intended contract. A
loss-making allocation cannot be an equilibrium allocation as the number of firms
n < ∞ such that some firm f that makes negative profits would withdraw contracts.
Proof of Proposition 3
Assume to the contrary that an equilibrium with an allocation
H
L
(w̄ H , w̄ L ) 6= (ωW
M S , ωW M S ), where no contract was withdrawn on the equilibrium
path, exists. We will show that there is a profitable deviation in stage 1 and hence a
contradiction.
Since it is an equilibrium, each firm makes a nonnegative expected profit. Since
(w̄ H , w̄ L ) is the equilibrium allocation and no contract
on the equilibS f is withdrawn
rium path, there does not exist a contract ω ∈
Ω0 with uJ (ω) > uJ (w̄ J ).
f ∈F
Now since (w̄ , w̄ ) 6=
there exist some firm f˜ and incentive compat
˜
ible contracts ω̃ H and ω̃ with u (ω̃ ) > uL (w̄ L ) such that if Ωft̂ = ω̃ H , ω̃ L and all
consumers buy their respective contract at firm f˜ in stage 3, the profit of firm f˜ would
be higher than the profit of firm f˜ when the equilibrium allocation is (w̄ H , w̄ L ): For
the argument when (w̄ H , w̄ L ) is not second-best efficient, see e.g. proof of Proposition
2.3 in Dosis (2016). When (w̄ H , w̄ L ) is second-best efficient, which implies zero profits
L
on the whole population, then uL (w̄ L ) < uL (ωW
M S ) and then it is easy to show that
there exists a menu that is strictly preferred by L-types and makes positive profits
on the whole population.
ˆ
H
L
t is a profitable deviaBut then, for some firm f˜, setting Ωft = {ω̃
S , ω̃f } for all
J
tion: First, since there is no contract ω ∈
Ω0 with u (ω) > uJ (w̄ J ), and since
H
L
H
L
(ωW
M S , ωW M S )
L
L
L
f ∈F \f˜
L
L
L
L
u (ω̃ ) > u (w̄ ), withdrawal of some contract(s) in some round t by some other
firm cannot induce L-types to not buy ω̃ L in stage 3 since ω̃ L always remains the
best contract on offer for L-types. Second, keeping {ω̃ H , ω̃ L } on offer for all t yields
firm f˜ higher profits than when the equilibrium allocation is (w̄ H , w̄ L ) and thus nonnegative profits such that the menu will not be withdrawn by firm f˜. Hence, there
is a profitable deviation and thus a contradiction.
Proof of Proposition 4.
“⇐” Fix an equilibrium in the original game with equilibrium strategy αt∗f (ht ). Then
consider the following strategy ᾱtf (ht ) in the extended game: Firm f ∈ F offers Ω∗f
0
in stage 1. In stage 2, round t = 1, the strategy specifies ᾱtf (ht ) = αt∗f (ht ). For
t ≥ 2, ᾱtf (ht ) = αt∗f (ht ) if Ωjt−1 ⊆ Ωjt−2 ∀j ∈ F . i.e., firms play as in the original
game as long as no contract was added by any firm in the previous round for t ≥ 2.
If, however, any firm fˆ 6= f adds a contract in t − 1, i.e. Ωjt−1 * Ωjt−2 for some
j ∈ F \ {f }, then ᾱtf (ht ) = Ω∗f
0 . I.e., if some firm adds a contract in some t − 1, then
all other firms reoffer stage 1 contract menus in t.
Now firstly, this strategy yields the same equilibrium allocation as in the original
game as on the equilibrium path, each firm f ∈ F takes the same action in stage 1
33
and in all rounds of stage 2 as on the corresponding equilibrium path in the original
game.
It remains to show that there is no profitable deviation. A profitable deviation here
means a deviation such that profits are higher than in equilibrium in the original
game. There are two cases to consider:
Case 1: Stage 2 does not end.
If stage 2 does not end, all firms make zero profits. Then a deviation is not profitable.
Case 2: Stage 2 ends.
Then, there exists a largest t where a deviator adds a contract, denoted by t̃. According to the above strategy, all non-deviating firms will then offer their initial set
of contracts in t̃ + 1. Then however, there is no profitable deviation from the proof
of Proposition 2.
“⇒” We prove this by contraposition. An allocation that is not an equilibrium allocation in the withdrawal game is not individually rational for consumers, incentive
compatible or loss-making. If the allocation is not individually rational or incentive
compatible, some type does not take her intended equilibrium contract. A lossmaking allocation cannot be an equilibrium allocation as the number of firms n < ∞
such that some firm f that makes negative profits would withdraw loss-making contract(s).
B Example n-type case
For the n-type case, MWS contracts are such that there might be several subgroups
with cross-subsidization within the subgroups, insurers exactly break even on the
subgroups and of course on all types jointly.36 The construction of MWS contracts
is done recursively: First, the expected utility of the highest risk type is maximized
subject to a non-negative profit constraint on this type. This gives a reservation utility for the highest risk type, which is the expected utility of its RS contract. For the
next highest risk type, the expected utility is maximized, subject to incentive compatibility for the higher risk type and a constraint that the higher risk type receives
at least the reservation utility as determined before, as well as an overall zero profit
constraint on these two types. This is the program that we show in section 3. The
expected utility for the second highest risk type from the solution of this program
is then the reservation utility of the second highest risk type in the program for the
third highest risk type, where the zero profit constraint is then on the three highest
risk types, and so forth. Thus, the program for the n-th type is maximizing the
expected utility of the nth- type, subject to incentive compatibility constraints, the
constraints that every type receives at least their reservation utility as determined in
the respective programs for these types, and an overall zero profit constraint. The
solution to this program are the n-type MWS contracts. An important observation is
that by construction of the MWS contracts for the n-type case, a deviation aimed at
pooling of risk types (where some types are cross-subsidized) is not profitable from
the outset. This is because if there were a profitable pooling contract (menu) in
which some type is cross-subsidized, it would already be incorporated into the MWS
36
See e.g. Spence (1978) or Picard (2015).
34
contracts. Thus, a deviation has to take the form of cream skimming to attract some
intermediate risk type(s) on which profits are made (separately) with MWS contracts.
Then, with strategies similar to the ones proposed in this paper, namely to withdraw
contracts which are unprofitable, and to offer a series of latent contracts for each risk
type to retaliate any deviator, i.e. to cream skim from the deviator, those deviations
can be made unprofitable.
w2
MWS contracts
u5
1
LR contracts for type 1
2
LR contracts for type 2
3
LR contracts for type 3
4
LR contracts for type 4
4
3
2
RS contracts
1
u4
u3
u2
w1
Figure 4: Equilibrium contract offers with 5 types
Figure 4 provides an illustration for the case of 5 types, following the example provided by Picard (2015). Type 5 is the highest risk type, type 4 the second highest
etc. Figure 4 shows the case that MWS contracts are such that the two highest risk
types, type 5 and type 4, are cross-subsidized by type 3, whereas the two lowest risk
types 2 and 1 receive individually zero profit making contracts. To prevent deviations, LR contracts are offered on every type except the highest risk type. For type
3, the cross-subsidizing type, LR contracts are constructed as those for the low risks
in the two-type case. For the types where the MWS contract is zero profit making
(types 1 and 2 in the example), the LR contracts are all contracts on and between the
MWS and RS contracts. Type 4, a cross-subsidized type who is not the highest risk,
LR contracts are all contracts on this type’s fair insurance line below the indifference
curve of the higher risk type 5 to the RS contract.
35
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