Universität des Saarlandes
University of Saarland
Volkswirtschaftliche Reihe
Economic Series
UNCERTAINTY AND STRATEGIC INTERACTION
IN ECONOMICS
Jürgen EICHBERGER and David KELSEY*
Economic Series No. 9914
December 1999
University of Saarland
Department of Economics (FB 2)
P.O. Box 151 150
D-66041 SAARBRUECKEN, GERMANY
Phone: +49-681-302 -4822, Fax:-4823
http://www.uni-sb.de/rewi/fb2/eichberger/
E-mail: [email protected]
Abstract
The endogeneity of equilibriumstrategies makes modelling uncertainty about the behaviour of
other economic players difficult. Recent developments in decision and game theory offer an opportunity to include strategic uncertainty as an explanatory variable in economic analysis.
This paper introduces to a new concept of decision making under uncertainty and an adapted notion of strategic equilibrium. With these tools, several well-known economic models of strategic
interaction are re-examined. These applications provide an opportunity to assess the contribution
of uncertainty for a better understanding of strategic interactionin economics.
JEL Codes: C72, D43, D62, D81
Keywords: Choquet expected utility, strategic complements and substitutes, economic
models
*
Research supported by ESRC grant no. R000222597. We would like to acknowledge helpful comments from
seminar participants at the Australian National University, the University of Alabama, the Humboldt Universität,
Berlin, and from participants of the meetings of the Royal Economic Society and the European Economic Association. Special thanks go to Sujoy Mukerji who instigated our interest in this topic and to Jeffrey Kline who pointed
out a mistake in an earlier version of the paper. Jeffrey Kline’s and Simon Grant’s comments on our discussion of
the irrational types analysis helped to clarify the exposition of this issue. We would also like to thank Peter Sinclair
and Willy Spanjers for their comments on the final version. The usual disclaimer applies.
1. Introduction
1.1
Motivation
Many economic theories of choice under uncertainty assume that any available course of
action gives rise to specified outcomes with known probabilities. However in practice it
is rare for the probabilities to be known when an economic decision needs to be made.
There are a few economic decisions where probabilities are known, for instance some
types of gambling and the sale of insurance, where the insurance company has access
to good actuarial tables. However we believe that such examples are relatively rare and
unknown probabilities are typical.
We believe that there are many important economic decisions where individuals are not
able to assign a probability distribution to the relevant variables or lack confidence in the
probability distribution if one is available. This may be due to the unfamiliarity of the
situation in which the decision-maker is placed or to the complexity of the operation of
assigning probabilities to events. We shall describe situations in which probabilities are
known or easily calculated as ‘‘risk’’and shall reserve the term ‘‘uncertainty’’to refer to
situations where probabilities are unknown. What we refer to as ‘‘uncertainty’’is sometimes called Knightian uncertainty or ambiguity. Recent years have seen a number of
advances in decision theory which enable us to model uncertainty in this sense.
In the present paper we aim to study the economic implications of these theories. We
identify economic effects of uncertainty by studying comparative statics. These show that
uncertainty has different effects depending on whether the actions of different economic
agents are strategic complements or substitutes. Assume that there are positive externalities between agents. Then if there are strategic substitutes/complements uncertainty
tends to increase/decrease the equilibrium action. These effects are reversed if there are
negative externalities. For symmetric games with strategic substitutes, equilibria with uncertainty are typically closer to the ex-post Pareto optimum than the conventional (Nash)
equilibrium.
A brief summary of our conclusions is that uncertainty tends to be helpful when there are
strategic substitutes but can have negative effects when here are strategic complementarities. We consider applications including the familiar Bertrand and Cournot oligopoly
models. We show that increases in uncertainty make Cournot models less competitive but
increase the intensity of Bertrand competition.
1.2
Knightian Uncertainty
Uncertainty has long been recognized as an important factor determining economic activities. Knight (1921) made a distinction between risk, i.e., situations where the probabilities are known, and uncertainty, i.e. situations where this is not the case. He used
this to explain economic phenomena such as profit and entrepreneurial activity. For several decades, however, the behaviourist theory of subjective expected utility by Savage
(1954) appeared to have rendered this distinction obsolete. If individuals faced with uncertainty behave as if they held beliefs that can be represented by a subjective probability
distribution over events, then, from an analytical point of view, behaviour under risk and
2
under uncertainty can be treated in the same way.
Yet early evidence by Ellsberg (1961) failed to support the hypothesis that beliefs
could be represented by conventional subjective probabilities. Systematic laboratory experiments have confirmed Ellsberg’s conjecture (Camerer and Weber, 1992). Certain aspects of uncertainty, in particular, partial information about the likelihood of events
cannot be properly incorporated in subjective probabilities. Another reason for being
sceptical about Savage’s theory is that it does not seem to be able to handle situations
where a decision-maker has a subjective probability but lacks confidence in it. Finally
we note that the Savage theory has difficulties in dealing with choices where some of the
outcomes may be unknown.1
Despite such obvious inconsistencies, expected utility theory proved to be a successful
modelling tool. Important economic insights were obtained from the distinction between
risk preferences and beliefs, which can be made in this approach. The economics of insurance and information could be developed in this context. Short of an alternative theory
of decision making under uncertainty, it was impossible to pursue aspects of uncertainty,
which subjective probabilities could not capture properly.
In recent years, substantial progress has been made modelling decision-making under uncertainty without subjective probabilities. Schmeidler (1989) and Gilboa (1987)
proposed a theory where a decision maker’s beliefs are represented by non-additive probabilities (or capacities ). Choquet expected utility (CEU) theory, a generalization of subjective expected utility, can accommodate non-additive probabilities, while maintaining
the separation of beliefs and outcome evaluation, which is important in economic applications.
1.3
Games with Uncertainty
More recently, this approach has also been applied to interactive decision theory where
uncertainty concerns the behaviour of other economic agents (Dow and Werlang,
1994, Marinacci 1996, Eichberger and Kelsey, 1999b). Behaviour and beliefs of agents in games are endogenously determined in equilibrium. Therefore applying
this new decision theory to uncertainty about this interaction requires a re-examination
of equilibrium concepts. Traditionally, equilibrium has been characterized by three assumptions:
² strategy sets and pay-offs are common knowledge,
² opponents are assumed to be rational in the sense of maximizing their objective functions by independently choosing a strategy,
² beliefs about opponents’behaviour have to be consistent with actual behaviour.
The new approaches no longer describe beliefs by conventional (additive) probability
distributions. Hence, even in equilibrium, beliefs can no longer simply be identified with
actual behaviour. Less stringent consistency requirements on the relation between beliefs
and behaviour have to be imposed. One can interpret this as a theory of boundedly rational
behaviour.
1
Mukerji (1997) argues that the models we use in the present paper can be adapted to model some
situations of decision-making with unforeseen outcomes
3
With non-additive beliefs it is easier to accommodate such empirically observed anomalies as, the Ellsberg-paradox (Ellsberg, 1961). Since a general capacity has a large
number of free parameters, predictions about behaviour however become much harder
without imposing reasonable restrictions on capacities. In this paper, we will restrict attention to beliefs that can be represented by simple capacities. This restriction reduces the
number of free parameters. Moreover, with simple capacities, one can give uncertainty
and confidence a parametric interpretation. This allows us to conduct comparative static
analysis with respect to exogenous changes in uncertainty.
Working with Choquet expected utility and simple capacities is almost as easy as with
expected utility theory. The theory is intuitive and can explain some puzzles of the traditional theory. More importantly however, it allows us to study the impact of uncertainty
and confidence on economic outcomes, an analysis which is impossible in the context
of expected utility theory. A player’s uncertainty may reflect his/her familiarity with a
particular situation. Despite such advantages, we believe that the success of the new
equilibrium concept will ultimately depend on its success in providing new explanations
for economic phenomena. In order to illustrate the potential of this approach, we will apply it to some standard economic models. These applications show that uncertainty has a
substantial impact on the equilibria of Cournot and Bertrand duopoly models. Uncertainty
has also important implications for the theory of public goods and externalities.
1.4
Organisation of the Paper
The following section introduces the concept of a capacity and the Choquet integral. Section 3 applies our decision theory to strategic games. A modified solution concept, equilibrium under uncertainty, is introduced. We will show that the impact of uncertainty
on the equilibrium depends on whether the strategies are strategic substitutes or complements. The general comparative static analysis is carried out in section 4, while section
5 presents economic examples which illustrate the potential of the approach to generate new results. In section 6, we will argue that uncertainty is a phenomenon which is
distinctly different from risk aversion and from games of incomplete information with
‘‘crazy’’types. All proofs are gathered in an appendix.
2. Choquet Integral of a Simple Capacity
In this section we present the concepts of a capacity and of the Choquet integral, an
expected value of a function with respect to a capacity. In addition, simple capacities
are introduced as a special case, and a representation of the Choquet integral for simple
capacities is derived.
Consider an economic agent whose profit may depend in part on the behaviour of rivals.
Let X µ Rm denote the set of actions rivals may take. Let X denote the set of events,
which are subsets of X: 2 We shall represent individuals beliefs by capacities which can
be viewed as non-additive subjective probabilities. A capacity is a set-valued function
representing the beliefs about events.
2
Formally we require that X is an algebra:
4
Definition 2.1 capacity
Let X µ Rm : A capacity is a function º : X ! R which assigns real numbers to events,
such that
(i) A; B 2 X; A µ B implies º(A) 6 º(B); monotonicity
normalisation
(ii) º(;) = 0 and º(X) = 1:
The normalisation condition is the same as for conventional probability distributions. It is
imposed for convenience only. Monotonicity is a weak consistency condition. It allows,
e.g., for set-valued functions where the value of a set plus the value of its complement
exceed one. It is therefore reasonable to restrict the class of capacities by excluding this
possibility. In particular, it appears natural to include (additive) probability distributions
as a special case. For the interpretation as beliefs, the following two extreme examples
of capacities play a special role.
Special cases:
1. A capacity is a probability distribution if, in addition, for all A; B 2 X, A \ B = ;;
º(A [ B) = º(A) + º(B)
holds.
2. The complete uncertainty capacity À is defined by
À(E) = 0 for all E $ X:
Complete uncertainty characterises a decision maker who is unable to assign likelihood to
any event. Probability distributions, at the other extreme, represent very precise beliefs
about the likelihood of events.
In this paper, acts are real-valued functions which associate outcomes to states. Since
economic examples deal naturally with quantities, prices, and values, the restriction to
state-contingent variables with real numbers as values is often adequate. As in standard
decision theory, one wishes to assign an expected value to acts, which a decision maker
may choose. The Choquet integral is similar to an expectation, but allows a cautious
response to uncertainty by over-weighting bad outcomes.
Let F be a set of functions f : X ! R; which represent the decision makers’pay-offs as
a function of the actions of his/her opponents.3 A weighted average of a function f 2 F
with respect to a capacity º can be defined by the Choquet integral. This is the analogue
of taking an expectation with respect to a probability distribution. Schmeidler (1989),
Gilboa (1987), and Sarin and Wakker (1992) have axiomatised maximising a
Choquet integral as a representation of preferences.
3
Formally we require that the set F consist of bounded and measurable real-valued functions on X. (A
function f : X ! R; is measurable if the sets fx 2 Xj f (x) ¸ tg and fx 2 Xj f (x) > tg are elements
of X for all t 2 R):
5
Definition 2.2 Choquet integral4
For any f 2 F ; the Choquet integral with respect to the capacity º is defined as
1
R
R
R0
f dº =
º(fx 2 Xj f (x) > tg) dt+
[ º(fx 2 Xj f(x) > tg) ¡ 1] dt:
¡1
0
In our opinion there are two features of the general CEU model, which make it difficult
to apply. Firstly it can be mathematically complex. Secondly there are too many free
parameters. A capacity on a set with n elements involves 2n parameters, while n ¡ 1
parameters will describe a probability distribution on the same set. It is therefore desirable to restrict attention to a smaller class of capacities. In Eichberger and Kelsey
(1999a) we propose several desiderata for capacities to be used in economic applications and identify the class of capacities satisfying them. Simple capacities belong to this
class. A simple capacity is a convex combination of an additive probability distribution
and the capacity of complete ignorance. Simple capacities maintain many properties of
additive probability distributions and have a natural interpretation in terms of beliefs. The
deviation from additivity ref lects the degree of uncertainty about the prediction made in
the probabilistic part. In games, one can determine the additive part of beliefs endogenously as the rational prediction of the players from the knowledge of the game structure
and the assumed rationality of others. Players may however distrust these ‘‘rational’’predictions to some degree. If they do not trust their prediction at all, they are in a state of
complete uncertainty.
Definition 2.3 simple capacity
A simple capacity based on an additive probability distribution ¼ is defined as
º(E) = ° ¢ ¼(E) + ° ¢ À(E)
for all E µ X with ° := 1 ¡ °:
Simple capacities can be interpreted as additive beliefs, which are held with some degree
of doubt. The degree of confidence ° 2 [0; 1] measures the confidence of the decision
maker in the probability ¼. In contrast, the weight given to the capacity of complete uncertainty, °; denotes the degree of uncertainty. In this interpretation, uncertainty is the
counterpart of confidence in a probabilistic assessment. In the standard system of representing beliefs by additive probabilities it is not possible to model the decision-maker’s
confidence in a probability assessment. In our opinion, this is a significant advantage of
using non-additive beliefs.
Various measures of uncertainty have been suggested in the literature5. Uncertainty is usually measured by the ‘‘deviation’’of a capacity from an additive probability distribution.
Since there are different notions of ‘‘deviation’’from additivity which are not compatible,
comparative static results in regard to players’confidence or uncertainty are difficult. In
contrast it is relatively easy to derive comparative static results with simple capacities.
4
We do not illustrate this general formula of the Choquet integral by examples since most of this paper
will be concerned with a special case of it which is easy to interprete and apply.
5
Dow and Werlang (1992), Marinacci (1996), Ghirardato and Marinacci (1998),
and Epstein (1999) suggest different measures of uncertainty and uncertainty aversion.
6
In economic applications, it is particularly useful that the Choquet expected value of a
function with respect to a simple capacity takes the form of a convex combination of the
expected utility with respect to the additive part ¼ and the worst outcome, weighted with
° and ° respectively. Eichberger and Kelsey (1999b) provide an axiomatisation
for this representation.
Proposition 2.1 Choquet integral of a simple capacity
Consider a simple capacity º = °¢¼+° ¢À where ¼ is an additive probability distribution
on a compact set X: The Choquet integral of a continuous function f on X has the
following form:
Z
Z
f dº := ° ¢ f d¼ + °¢ min f (x):
x2X
3. Strategic Games with Uncertainty
We will restrict this exposition to games with a finite set of players, whose beliefs can
be represented by simple capacities.6 Eichberger and Kelsey (1999b) extend
the analysis to games in which players’beliefs are represented by general capacities. In
economic applications, players’ strategy sets are mostly continuous variables, such as
prices, quantities and investment expenditures. For this reason, we will focus here on
equilibria under uncertainty in pure strategies7.
Consider a game
¡ = (I; (Si ; pi )i2I )
with
² a finite player set I = f1; 2; :::; Ig;
² strategy sets Si µ Rmi ; and
² payoff functions pi (si ; s¡i ):
The following notational conventions will be maintained throughout this paper. The set
of strategy combinations will be denoted by S = S1 £ ::: £ SI : A typical strategy combination s 2 S can be decomposed into the strategy (component) si of player i and the
strategy combination of the opposing players s¡i ; i.e., s = (si ; s¡i ): The set of strategy
combinations of all players except player i is denoted by S¡i :
Each player i is assumed to choose the best strategy available, si 2 Si ; given the beliefs
about the other players’strategy choices, s¡i : In a pure strategy equilibrium, beliefs are
concentrated on a single profile of strategies of the opponents. One can interpret such beliefs as rational predictions of the opponents’actions about which the player is uncertain.
Though it is not modelled explicitly, such uncertainty may be due to doubts about whether
6
This assumption is made to simplify the analysis and to make the intuition for our results more clear.
We would view it as analogous to assuming that a production function has the Cobb-Douglas form. Related
results where this assumption is not imposed can be found in Eichberger and Kelsey (1999c).
7
Finite strategy sets are treated in Dow and Werlang (1994) and in Eichberger and Kelsey
(1999b).
7
one’s information on the preferences or the rationality of the opponents is correct.8
Denote the probability distribution concentrated on s¡i 2 S¡i ; by ±(¢js¡i ) i.e.:
½
1 for s¡i 2 E
±(Ejs¡i ) =
; for all E µ S¡i ;
0 for s¡i 2
=E
and consider the simple capacity º i (¢js¡i ) concentrated on this strategy combination:
º i (Ejs¡i ) := ° i ¢ ±(Ejs¡i ) + ° i ¢ À(E); for all E µ S¡i :
This capacity º i (¢js¡i ) represents the beliefs of player i regarding the strategy choice of
the other players s¡i . The parameter ° i reflects the confidence of player i in the prediction
±(¢js¡i ): Although beliefs are concentrated on a single profile of strategies, the player is
not completely confident about this belief.
We shall now define an equilibrium of a game when there is Knightian uncertainty about
the behaviour of other players.
Definition 3.1 An Equilibrium under Uncertainty
is a profile of simple capacities
R
(º i (¢js¤¡i ))i2I such that s¤i 2 argmaxsi 2S i pi (si ; s¡i ) dº i (¢js¡i )
= argmaxsi 2S i [° i ¢ pi (si ; s¡i ) + (1 ¡ ° i )¢ min pi (si ; s¡i )]; for 1 6 i 6 I:9
s¡i 2S¡i
In an equilibrium, players are supposed to choose a best response given their beliefs about
the opponents’behaviour. The Choquet expected payoff function of a given player incorporates the uncertainty about the opponents’ predicted behaviour. In predicting the
behaviour of others, the player takes their own uncertainty into account. In equilibrium
the prediction coincides with actual behaviour. Hence, the uncertainty of all players inf luences the equilibrium. Note that in an equilibrium under uncertainty, the beliefs of
any given player are concentrated on a single strategy profile of his/her opponents. Thus
equilibrium under uncertainty is a generalisation of Nash equilibrium in pure strategies.
Since the strategy space is convex, we do not need to consider mixed strategies. This is
an advantage, since mixed strategies are difficult to interpret when payers have additive
beliefs. These difficulties are increased when beliefs are non-additive.
Proposition 3.1 shows that an equilibrium under uncertainty can be interpreted as a Nash
equilibrium of a perturbed game. The result follows from the proof of Proposition 3.1 in
Eichbergr and Kelsey (1999b).10
R
Definition 3.2 Define Pi (si ; s¡i ; ° i ) := pi (si ; s¡i ) dº i (¢js¡i ):
Thus Pi denotes player i’s expected pay-off given (s)he has beliefs given
R by a simple capacity concentrated at s¡i : Proposition 2.1 implies Pi (si ; s¡i ; ° i ) = ° i ¢ pi (si ; s¡i ) d±(¢js¡i )+
8
The interpretation of this model in terms of possible irrationality of the opponents is explored in more
detail in Rothe (1999).
9
The Equilibrium under Uncertainty in is a straightforward adaptation of the concept of a Nash Equilibrium under Uncertainty proposed by Dow and Werlang (1994). In Eichberger and Kelsey
(1999b) we provide a general definition of an Equilibrium under Uncertainty for beliefs represented by
arbitrary convex capacities. Dow and Werlang considered games with finite strategy sets. The definition
here is the natural extension of this solution concept to games with continuous strategy spaces.
10
In Eichbereger and Kelsey (1999b) the result is proved for games with finite strategy spaces.
The extension to games with continuous strategy spaces is straightforward.
8
(1 ¡ ° i )¢ min pi (si ; s¡i ) = ° i ¢ pi (si ; s¡i ) + (1 ¡ ° i )¢ min pi (si ; s¡i ):
s¡i 2S¡i
s¡i 2S¡i
- ¡
¢
¡
¢®
Proposition 3.1 The profile of simple capacities º 1 ¢js¤¡1 ; :::; º I ¢js¤¡I is an Equilibrium under Uncertainty if and only if hs¤1; :::; s¤1 i is a pure strategy Nash equilibrium
of the game
¡(° 1 ; :::; ° I ) = (I; (Si ; Pi (¢; ° i ))i2I ):
Note that an Equilibrium under Uncertainty of the game ¡(1; :::; 1) is a Nash equilibrium
of the game ¡: This result enables us to apply many well-known methods of analysis
in order to derive results in applications. The economic impact of uncertainty becomes
transparent because, for simple capacities, the deviation from additivity can be easily
measured.
We show below that Equilibria under Uncertainty in pure strategies exist under standard
assumptions.
Proposition 3.2 If, for all players i 2 I;
(i) the strategy sets Si are compact and convex, and if
(ii) the payo¤ functions pi (si ; s¡i ) are continuous in s and quasi-concave in each player’s own strategy si , then there exists an Equilibrium under Uncertainty in pure strategies.
4. The Impact of Strategic Uncertainty
In this section, we will study how changes in uncertainty affect the best responses and the
equilibrium. Uncertainty of a player about his/her opponents’behaviour can be described
by the degree of uncertainty ° i = 1 ¡ ° i ; i 2 I. Equivalently, one can use the degree of
confidence in the equilibrium prediction, ° i : The higher ° i (equivalently, the lower ° i ),
the more uncertain is player i about the opponents’strategy choices.
Since we deal with arbitrary finite numbers of players, the following notational conventions are important. For any two vectors x; y 2 Rn ; we write x > y if xk > yk for all
k = 1; :::n; and x > y if x > y and x 6= y: An interval of the vector space Rn is a
set fx 2 Rn j a 6 x 6 bg for some vectors a; b 2 Rn with a 6 b: We will make two
simplifying assumptions which are satisfied in many economic applications11 .
Assumption 4.1 (Strategic Complementarity) For all players i 2 I,
1. the strategy sets Si are intervals of R;
2. the payoff functions pi (si ; s¡i ) are twice continuously differentiable with
@ 2 pi (si ; s¡i )
>0
@si @sj
for all j 2 I; j 6= i:
It is not technically difficult to extend our results to strategy sets that are subsets of Rn (Milgrom
and Roberts, 1990, Fudenberg and Tirole 1991), but the assumptions for a game to be
supermodular are then harder to interpret in economic terms.
11
9
For two-player games, Bulow, Geneakoplos, and Klemperer (1985) introduced the notion of strategic substitutes and strategic complements which has become
widely used in industrial organisation and macroeconomics (compare Tirole, 1988 and
Cooper and John, 1988). Strategies are strategic substitutes (respectively, strategic
complements) if an increase in the activity of the opponent decreases (respectively, increases) the marginal payoff of a player. Because we allow for many rival players, the
following modified definition is used.
Definition 4.1 strategic complements and substitutes
Strategies are
strategic complements, if
and
² strategic substitutes,
if
²
@ 2 pi (si ;s¡i )
@si @sj
> 0;
@ 2 pi (si ;s¡i )
@si @sj
<0
for all si 2 Si ; all sj 2 Sj and all j 6= i holds.
A game is characterised by strategic complements ( strategic substitutes) if the strategies of all players are strategic complements (strategic substitutes).
The second part of Assumption 4.1 requires strategies to be strategic complements. In
general, this excludes games with strategic substitutes. For two-player games however,
one can redefine games with strategic substitutes as games with strategic complements
(Fudenberg and Tirole 1991, p. 492). Hence, Assumption 4.1 covers a large
number of economic applications.
Strategic uncertainty increases a player’s sensitivity to the worst actions which opponents
might choose. It is therefore important to know how the opponents’ strategy choices
affect a player’s payoff. Hence, Cooper and John (1988) distinguish games also
according to positive and negative spillovers.
Definition 4.2 positive and negative spillovers
A strategy of a rival player sj 2 Sj creates
² positive spillovers if
and
² negative spillovers if
@pi (si ;s¡i )
@sj
>0
@pi (si ;s¡i )
@sj
< 0:
A game is characterised by positive ( negative) spillovers if the strategies of all rivals
create positive (negative) spillovers for all players.
If a game has positive/negative spillovers, then the worst strategy profile which the other
players may choose is the lowest/highest available strategy. In particular the worst strategy profile is independent of the player’s own strategy. This simplifies the analysis of
comparative statics. In particular, if the game ¡ satisfies Assumption 4.1, then the induced game ¡(° 1 ; :::; ° I ) = (I; (Si ; Pi (¢; ° i ))i2I ) satisfies it also.
10
Lemma 4.1 Let ° i > 0 for all i 2 I: Given Assumption 4.1, for all i; j 2 I; i 6= j and
all si 2 Si ; sj 2 Sj ;
1. if a game is characterised by positive (negative) spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0;
@si @sj
2. if a game has positive spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0;
@si @° i
3. if a game has negative spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0:
@si @° i
Lemma 4.1 allows us to apply some established results about supermodular games. Milgrom and Roberts (1990) and Fudenberg and Tirole (1991, 489-499)
provide a good introduction to supermodular games. Though we will make use of some
of the results from this theory, we will not introduce the concept here.
We shall now examine how uncertainty affects the best responses of a given player. Denote
by
½i (s¡i ; ° i ) = arg max Pi (si ; s¡i ; ° i )
si 2Si
the set of best responses of player i: In the context of games with strategic complements,
the reaction function will shift monotonically. For positive spillovers, more uncertainty
of a player, ° i smaller than ° 0i ; will unambiguously reduce the optimal strategy level si :
For negative spillovers, more uncertainty means higher optimal strategy levels.
Proposition 4.1 Let si 2 ½i (s¡i ; ° i ) and s0i 2 ½i (s¡i ; ° 0i ): Given Assumption 4.1,
1. if a game has positive spillovers, then
° i > ° 0i implies si > s0i ;
2. if a game has negative spillovers, then
° i > ° 0i implies si 6 s0i :
Figure 1 illustrates this result for the case of a two-player game. For the case of twoplayer games, this result would suffice to analyse the equilibrium effects of a change in
uncertainty diagrammatically. Note however that Proposition 4.1 holds for general nplayer games.
For strategic complements, comparative static analysis is possible. Denote by ° :=
(° 1 ; :::; ° I ) the vector of the degrees of confidence of all players and by S ¤ (°) the set
of equilibrium strategy combinations s¤(°) of the game ¡(°) = (I; (Si ; Pi (¢; ° i ))i2I ):
Each such strategy combination is an Equilibrium under Uncertainty.
The following result implies monotonicity of the equilibrium strategies in °; if there is a
unique equilibrium. Moreover, for multiple equilibria it guarantees that the whole set of
equilibria moves in a particular direction. For strategic complements, the set of equilibria
11
is partially ordered and contains a greatest and a smallest element. If there are positive
spillovers, then the strategies in the extreme equilibria will fall with rising uncertainty,
while, for negative spillovers, they will rise. Note that these results do not rely on marginal
changes. Any monotonic change in uncertainty will induce monotonic changes of the
equilibrium strategy choices. Figure 1 illustrates the case of a unique equilibrium and the
case of three equilibria.
s2
½1 (s2.; °)
½1 (s2; °)
½1 (s2; ° 0 )
½2 (s1 ; ° 0 )
½2(s1 ; °)
s2
½1 (s2 ; ° 0 )
½2(s1 ; ° 02 )
½2 (s1; °)
s¤2(° 0 )...................................
..
s¤2 (°) ........................... ...
.. ..
.. ..
.. ..
... ...
.. ..
s1
0
s¤1(°) s¤1 (° 0 )
unique equilibrium
s1
0
multiple equilibria
Figure 1: Uncertainty and equilibria
Proposition 4.2 Given Assumption 4.1,
1. there exist s¤(°); s¤ (°) 2 S ¤(° 1 ; :::; ° I ) such that
s¤ (°) 6 s¤ (°) 6 s¤(°);
2. if a game has positive spillovers, then ° > ° 0 implies
s¤ (°) > s¤ (° 0 ) and s¤ (°) > s¤ (° 0 );
3. if a game has negative spillovers, then ° > ° 0 implies
s¤ (°) 6 s¤ (° 0 ) and s¤ (°) 6 s¤ (° 0 ):
The following subsections deal with two special cases which are important for economic
applications. Moreover, in these cases, one can apply most of the results of this section
also to the case of strategic substitutes.
4.1
Two-player Games
Two-player games are common in economic applications. From industrial organisation
to international trade, two-player games are often the only way to draw clear conclusions
about the effects of exogenous shifts in parameters. The following result shows that,
in two-player games, a straightforward modification of the game allows us to treat also
games with strategic substitutes as supermodular games. Reversing the order on one play12
er’s strategy set enables us to convert a game of strategic substitutes into one of strategic
complements.
Lemma 4.2 For ¡ = (f1; 2g; (Si ; pi )i=1;2) ; let Se1 := S1 ; Se2 := ¡S2; pe1 (e
s1 ; se2 ) :=
p1(e
s1 ; ¡e
s2); and pe2 (e
s1; e
s2 ) := p2 (e
s1; ¡e
s2 ):
³
´
e = f1; 2g; (Sei ; pei )i=1;2
1. If strategies in ¡ are strategic substitutes, then strategies in ¡
are strategic complements.
2. Moreover, if ¡ has positive (negative ) spillovers then
² pe1 has negative (positive ) spillovers,
² pe2 has positive (negative ) spillovers.
Hence, for a two-player game with strategic substitutes ¡; where the strategy sets are
e is a supermodular game. Moreover, Lemma 4.2 allows us to apply the
intervals of R, ¡
results of Proposition 4.1 to games with strategic substitutes.
Proposition 4.3 Let si 2 ½i (sj ; ° i ) and s0i 2 ½i (sj ; ° 0i ); i 6= j; and suppose that strategies in ¡ are strategic substitutes.
1. If a game has positive spillovers, then
° i > ° 0i implies si 6 s0i :
2. If a game has negative spillovers, then
° i > ° 0i implies si > s0i :
Figure 2 illustrates how an increase in uncertainty, ° < ° 0 ; affects the best response
correspondence of a player.
Proposition 4.3 does not depend on marginal changes. It is much stronger than the usual
comparative static results. In contrast to the case of strategic complements, it is impossible
to derive a global result of equilibrium comparative statics similar to the one in Proposition
4.2. As Figure 3 illustrates, with strategic substitutes, even for a unique equilibrium, no
monotonicity of the equilibrium correspondence can follow from monotonicity of the
best response correspondences.
4.2
Symmetric games
A second group of games frequently used in economic applications are symmetric games,
i.e. games with identical players. For allocation problems with public goods or for common resource problems, it is a widespread practice to consider the case of an arbitrary
number of identical players. A game is symmetric if all players have the same strategy
set and utility function.
Definition 4.3 symmetric games
(a) A game ¡(° 1 ; :::; ° I ) is symmetric if, for all i; j 2 I; i 6= j;
² Si = S;
² pi (si ; sj ; s¡ij ) = pj (sj ; si ; s¡ij );
² ° i = °:
13
° decreases
si
6
positive spillovers
negative spillovers
?
½i (sj ; °)
sj
0
Figure 2: Strategic substitutes: increase of uncertainty
(b) An Equilibrium under Uncertainty in pure strategies is symmetric if, for all i 2 I;
s¤i = s¤:
It is not difficult to check that a symmetric game has a symmetric Equilibrium under
Uncertainty given the assumptions of Proposition 3.2.12
In this section, we will study the monotonicity properties of symmetric Equilibria under
Uncertainty. Assume therefore that all other players choose the same strategy y 2 S; and
denote by x 2 S the own strategy of the player. The payoff of any player can now be
written as
p(x; y) := pi (x; y; :::; y ):
| {z }
I¡1
Hence,
@ 2 p(x; y) X @ 2pi (x; y; :::; y)
@ 2 pi (x; y; :::; y)
=
= (I ¡ 1) ¢
; j 6= i:
@x@y
@x
@x
i @xj
i @xj
j6=i
Symmetric games have strategic substitutes (complements) if the original game has strategic substitutes (complements) as defined in Definition 4.1.
In order to study the impact of ambiguity, we consider the (Choquet) expected pay-off in
the symmetric game
P (x; y; °) := ° ¢ p(x; y) + (1 ¡ °)¢ min p(x; y):
y2S
12
This can be proved by suitably adapting the proof for existence of a symmetric Nash equilibrium in a
symmetric game, see Moulin (1986) p. 115.
14
s2
½1 (s2 ; ° 0 )
½1(s2 ; °)
s¤2 (°) . . . . . . . . . . . . . . . . . . . . . . ...s
..
..
..
..
..
.
s¤2 (° 0 ) . . . . . . . . . . . . . . . . . . . . . . ... . . . . . ...s
..
..
..
..
..
..
..
..
..
..
..
..
0
s¤1 (°) s¤1 (° 0 )
½2 (s1 ;0°)
½2 (s
1; ° )
s1
Figure 3: Strategic substitutes: equilibrium effects
Denote by ½(y; °) = arg max P (x; y; °) the best response correspondence. One can
x2S
modify the arguments of Propositions 4.1 and 4.3 in order to obtain the following result.
Proposition 4.4 Let x 2 ½(y; °) and x0 2 ½(y; ° 0 ): Suppose ° > ° 0 : If a game has
1. positive spillovers and strategic complements, or
negative spillovers and strategic substitutes, then
x > x0 ;
2. positive spillovers and strategic substitutes, or
negative spillovers and strategic complements, then
x 6 x0 :
If cross derivatives of payoff functions are strictly positive, as we have assumed throughout this section, one can obtain the stronger result that ° > ° 0 implies x > x0 if either
x or x0 is an interior equilibrium. For the case of strategic substitutes, there is a unique
symmetric equilibrium.
Proposition 4.5 If a symmetric game is characterised by strategic substitutes, then
there is a unique symmetric Equilibrium under Uncertainty.
The comparative static effects for the equilibrium strategies follow now directly from the
monotonicity of the best response correspondences (Proposition 4.4).
For strategic complements, uniqueness of a symmetric Equilibrium under Uncertainty
cannot be guaranteed. The Figure 4 illustrates how an increase in uncertainty, ° 0 < °;
15
affects the symmetric Equilibria under Uncertainty both for strategic substitutes and for
strategic complements if there are positive spillovers. The diagram shows the reaction
curve of a typical individual. By symmetry the equilibrium occurs where the reaction
curve crosses the 45o line.
x
° > °0
x
° > °0
½(y; °)
x0 ....................................r
..
..
¤ ......................r
..
x
½(y; ° 0 )
..
..
..
..
..
..
..
..
½(y; °)
..
.
y
0
x¤
x0
strategic substitutes
½(y; ° 0 )
y
0
strategic complements
Figure 4: Symmetric equilibria: increase in uncertainty
The right-hand figure shows the case where the increase in uncertainty leads to an decrease in the number of equilibria. Notice however that the highest and lowest equilibria
change monotonically.
5. Economic Applications
Applying the concept of an equilibrium under uncertainty to economic models reveals the
contribution of our model to a better understanding of these situations. Four examples
will be considered: two from industrial organization and two from the theory of externalities and public goods. In these cases, the analysis of the previous sections applies
without modifications. In these examples no attempt is made to provide the most general
framework for the results, since we have already presented general results in section 4.
This section rather tries to illustrate with simple and familiar examples the kind of results
which can be expected by our approach. We aim to show that uncertainty can be treated
without too much technical sophistication.
5.1
Quantity Duopoly
First we consider a Cournot-style duopoly model, where firms choose output levels. The
quantity duopoly case shows that uncertainty about rivals’ behaviour may reduce competition in a market because it induces less aggressive behaviour.
Consider a market for a homogeneous commodity in which n firms compete. Firms have
16
identical cost functions
c(xi ) = k ¢ xi ; k > 0:
Each firm chooses the quantity xi which it wants to supply from the interval [0; A]: We
assume a linear inverse demand curve, which gives the market price D as a function of
n
P
aggregate supply X =
xi :
i=1
D(X) = maxfa ¡ b ¢ X; 0g; A ¢ b > a > k > 0:
With these assumptions, firms are symmetric and we need to consider only the profit function of a representative firm. Moreover, we will focus on symmetric equilibria. Denote
by x the quantity supplied by the representative firm and by y the quantity supplied by
each of its competitors. The profits of the representative firm are given by:
¼(x; y) = maxfa ¡ b ¢ (x + (n ¡ 1) ¢ y); 0g ¢ x ¡ k ¢ x:
In order to determine its optimal strategy x, the firm has to predict the supply decision of
the other firms y: If beliefs are represented as simple capacities a firm over-weights the
worst outcome. We assume that a firm perceives the worst case to be a situation where
the opponents dump a large quantity on the market, which will drive the market price
down to zero. This is assumed here, min ¼(x; y) = ¡k ¢ x: One can, of course, make less
y
severe assumptions about the worst outcome from the opponents’quantity choices. Our
conclusions are robust to variations in this assumption since they would only affect the
strength of the impact which uncertainty has on a firm’s decision, not its direction. 13
Beliefs are modelled by a simple capacity concentrated on the opponents’output level y:
A typical firm’s (Choquet) expected payoff is given by.
P (x; y; °) = ° ¢ ¼(x; y) + (1 ¡ °)¢ min ¼(x; y)
y
= ° ¢ [a ¡ b ¢ (x + (n ¡ 1) ¢ y)] ¢ x ¡ k ¢ x;
where ° denotes the degree of confidence of the firm’s prediction of the opponents’behaviour y: The expected pay-off, P (x; y) is strictly concave in x and differentiable. Moreover,
the quantities supplied are strategic substitutes,
@ 2 ¼(x; y)
= ¡b ¢ (n ¡ 1) < 0;
@x@y
with negative spillovers,
@¼(x; y)
= ¡b ¢ (n ¡ 1) ¢ x < 0:
@y
It follows from Propositions 4.4 and 4.5 that an increase in uncertainty will,
² decrease the quantity supplied by each firm,
° > ° 0 implies x > x0 ;
13
If one allowed more general beliefs represented by a convex capacity which is not necessarily simple
then a number of bad outcomes would be over-weighted rather than just the worst outcome. This would
eliminate this sensitivity to choice of the worst outcome. We have not taken this route here since we believe
it would unduly increase the mathematical complexity of this paper. For examples of the application of
general capacities to related problems see Eichberger and Kelsey (1999c).
17
² raise the price in the market.
Figure 5 shows the best response function of the representative firm,
° ¢ a ¡ k (n ¡ 1)
½(y; °) = maxf
¡
¢ y; 0g:
2¢°¢b
2
x
° > °0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..s
..
..
..
..
..
..
¤ 0
..
x (° ) . . . . . . . . . . . . . . . . . . ...s
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
0
¤ 0
¤
x (° )
x (°)
x¤ (°)
½(y; °)
½(y; ° 0 )
y
Figure 5: Cournot equilibrium and uncertainty
Firms will offer a smaller quantity the more uncertainty they perceive. Uncertainty unambiguously reduces the amount brought to market. Intuitively, uncertainty makes a decision
maker cautious about the behaviour of his/her opponents. By dumping output onto the
market, the other firms can drive the price down. If firms become more concerned about
this possibility, they will reduce output in order to cut losses that would arise in such a
case. Notice that this analysis would remain qualitatively unchanged if, maybe more realistically, a firm perceives the worst possible behaviour of the opponent resulting in a
price which was strictly positive but below the Cournot level. Uncertainty, thus, reduces
competition. As our general results demonstrate, these conclusions do not depend critically on the particular cost and demand functions. They will hold for any pair of such
functions for which the quantity duopoly is a game of strategic substitutes.
Given the informational requirements of a market analysis, it will be difficult to identify
the Nash equilibrium quantities in an actual oligopolistic market. Hence, deviations may
be hard to measure. Indirect evidence may be gleaned from some experimental studies.
Though the experiment was designed to study learning behaviour in oligopoly situations,
Huck, Normann and Oechseler (1999) found that ‘‘more information about
behaviour and profits of others yields more competitive outcomes’’(Result 2, p. C89).
If uncertainty ref lects a lack of confidence in one’s information about the opponent, this
experimental result may provide some evidence for uncertainty as a reason for reduced
18
competition.
Traditionally, it has been suspected that oligopolies are prone to informal collusive arrangements. Scherer (1970) provides many examples (Chapter 6) from anti-trust cases. The
presumption of regulators that oligopolists collude, suggests that output is, at least sometimes, below the Cournot level without clear evidence of collusion. Strategic uncertainty
may offer an alternative and as yet unexplored explanation for why competition may be
less fierce in a quantity duopoly than predicted by Nash equilibrium.
5.2
Price Duopoly
In this section we investigate a model of price (Bertrand) competition. As our general
results predict, in this case, uncertainty has the opposite effect. More uncertainty increases
competition by inducing firms to charge lower prices.
Consider n firms producing heterogeneous goods which are close (but not perfect) substitutes. Firms have identical cost functions
c(xi ) = k ¢ xi ; k > 0;
where k denotes the marginal cost of production. Each firm is a local monopolist in its
market and charges the price pi for its output. It appears reasonable to assume that no firm
charges a price below its marginal cost k: Hence, the strategy set of a firm is the interval
[k; A]: We assume that firm i faces a linear demand curve given by:
X
d(pi ; p¡i ) = maxf0; a + b¢
pj ¡ c ¢ pi g; a; b; c > 0:
j6=i
Concentrating on symmetric Equilibria under Uncertainty, we denote by px the price
charged by a representative firm in its market. The other firms are assumed to charge
the price py in their markets. With these assumptions, one obtains the symmetric profit
function:
¼(px ; py ) := (px ¡ k) ¢ [a + b ¢ (n ¡ 1) ¢ py ¡ c ¢ px ]:
One easily checks that
@¼(px ; py )
= b ¢ (n ¡ 1) ¢ (px ¡ k) > 0
@py
and
@ 2 ¼(px ; py )
= b ¢ (n ¡ 1) > 0
@px @py
holds. Thus, this is a game with strategic complements and positive spillovers. Proposition 4.2 implies that the highest and the lowest of the equilibrium prices will be decreasing with increasing uncertainty.
To gain more insight, consider the best response function of a representative firm. The
worst case arises if all competitors lower their prices to their marginal cost,
min ¼(px ; py ) = (px ¡ k) ¢ [a + b ¢ (n ¡ 1) ¢ k ¡ c ¢ px ]:
py
As in the previous section our results would continue to hold under different assumptions
about what firms perceive to be the worst possible behaviour of their rivals. If beliefs
about the opponents’price strategy py are uncertain with a degree of confidence °, the
19
following Choquet expected profit function results:
P (px ; py ) = ° ¢ ¼(px ; py ) + (1 ¡ °)¢ min ¼(px ; py )
py
:
= (px ¡ k) ¢ (a + b ¢ (n ¡ 1) ¢ [° ¢ py + (1 ¡ °) ¢ k] ¡ c ¢ px ):
Optimization with respect to px yields the best reply function:
a + c ¢ k + b ¢ (n ¡ 1) ¢ [° ¢ py + (1 ¡ °) ¢ k]
;
½(py ; °) =
2¢c
which is depicted in Figure 6. Since the reaction function is increasing in °, an increase
in uncertainty ° = 1 ¡ ° will imply a lower price in any symmetric equilibrium under
uncertainty.
px
A ..............
½(py ; °)
p¤x (°) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...s
..
..
..
..
p¤x (° 0 ) . . . . . . . . . . . . . . . . . . . . ...s
..
..
..
..
..
..
..
.
.
.
.
.
.
.
.
.
.
.
.
.
k
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
0
k p¤x (° 0 ) p¤x (°)
½(py ; ° 0 )
..
..
..
..
..
.
A
py
Figure 6: Bertrand equilibrium and uncertainty
In contrast to the Cournot case, heterogeneous duopolists have their own markets in which
to react to the other firm’s price. Uncertainty about the other firm’s price amounts to
uncertainty about a firm’s own demand. The lower a given firm sets the price, the smaller
the market the opponents will face. Firms’concern about low demand in their respective
market provides an incentive for charging lower prices than in a conventional (Bertrand)
equilibrium.
5.3
Common Resource
In the third example we consider a fairly standard commons problem. There are two
firms who use a common resource as an input. We model this as a game where each
firm’s strategy is the amount of resource it extracts from the commons.
Consider two countries whose fisheries harvest the same part of an ocean. Both fisheries sell their catch, x and y respectively, on the world market at a common price q: For
20
concreteness, assume the following cost function:
c(x; y) = a ¢ (x + y) ¢ x:
This function captures the idea that increasing the output of one country increases total
and marginal cost of the other. Under these assumptions, the commons problem is a game
of strategic substitutes. Denote by x the maximal output of a country’s fishery, then a
strategic game is given by the
² strategy set
X = [0; x]
and the
² payoff function p(x; y) := q ¢ x ¡ a ¢ (x + y) ¢ x:
If players are uncertainty averse with confidence parameter °, then one obtains the following Choquet integral for a simple capacity concentrated on y :
P (x; y) := q ¢ x ¡ a ¢ (x + [° ¢ y + (1 ¡ °) ¢ x]) ¢ x:
As best reply function one obtains
q
° ¢ y + (1 ¡ °) ¢ x
½(y; °) = maxf0;
¡
g;
2¢a
2
which is illustrated in Figure 7 for two values of °:
x
° > °0
45o
.
½(y; °)
. . . . . . . . . . . . . . . . . . . ..s
x¤ (°) . ....................
.
¤ 0 . . . . . . . . . . . . . . . . .s ..
x (° )
.. ..
.. ..
.. ..
.. ..
.. ..
.. .. ½(y; ° 0 )
.. ..
0
x¤ (° 0 ) x¤ (°)
y
Figure 7: Common ressource
The output of the two fisheries declines as they become more uncertain about the others
behaviour. Fear of a false prediction induces both fisheries to lower their exploitation of
the common resource.
In this example, uncertainty mitigates the over-exploitation of the resource in a Nash
equilibrium. As before this result does not depend on the particular functional forms but
21
will apply whenever exploitation of a common resource may be described as a game of
strategic substitutes.
5.4
Weaker-link Public Goods
Public good problems provide a wide field of applications for the study of the impact of
uncertainty on equilibria. Depending on the technology of the public good and on the
utility function of the consumers, the strategic interaction arising may be characterised
by strategic complements or substitutes. Eichberger and Kelsey (1999c) study
Equilibria under Uncertainty in a general model with finite strategy sets. In the context of
continuous strategy sets, a public good economy with a weaker-link production function14
provides an example for strategic complements with multiple equilibria. This production
function implies that an individual will get more benefit from his/her own contribution to
the public good, the higher are the contributions of others.
Consider a technology which allows n players to produce a public good from contributions xi with increasing returns to scale according to the technology:
Q
f(xi ; x¡i ) = a ¢ xi ¢
xj :
j6=i
All players are endowed with one unit of funds. Hence, the strategy sets are the unit
interval, xi 2 [0; 1]: The opportunity cost of contributions is given by a quadratic cost
function
c
c(xi ) = k ¢ xi + ¢ x2i :
2
Denote again the contribution of a representative player by x and the contribution of each
opponent by y; then one obtains the following payoff function:
c
p(x; y) := a ¢ x ¢ yn¡1 ¡ k ¢ x ¡ ¢ x2 :
2
Notice that the contribution of the other players y has a positive impact not only on the
payoff level but also on the marginal product of x: Hence, we deal with a game of strategic complements and positive spillovers. Proposition 4.2 can be applied to show that
increasing uncertainty, ° > ° 0 ; implies decreasing contributions, x > x0 :
The worst case for a player arises if opponents do not contribute at all. In this case, the
output of the public good falls to zero and the player suffers the cost of the contribution
without any benefit. Given uncertainty of the players represented by a simple capacity,
one obtains the following Choquet integral,
c
P (x; y; °) := ° ¢ a ¢ x ¢ y n¡1 ¡ k ¢ x ¡ ¢ x2 :
2
Straightforward optimization with respect to x yields the best response function:
½
¾
° ¢ a ¢ yn¡1 ¡ k
½(y; °) := max 0; minf1;
g :
c
Figure 8 shows the best reply function for different values of °: For °; there are three
14
The classification of technologies follows Cornes and Sandler(1996, pp. 184-9). The reader is
referred to this book for further economic motivation.
22
x
½(y; °)
x¤¤ (°) = 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..s
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
s
.
x¤¤¤ (°) = x¤¤¤(° 0 ) = 1
x¤(°)
x¤ (°)
s
½(y; ° 0 )
y
x¤¤ (°) = 1
Figure 8: Weakest-link public goods and uncertainty
Equilibria under Uncertainty:
(i)
maximum contribution:
x¤¤ (°) = 1;
(ii) minimum contribution:
x¤¤¤ (°) = 0;
(iii) intermediate contribution: 0 < x¤ (°) < 1:
These equilibria can be ranked in the Pareto sense: the higher the contribution in equilibrium, the higher the equilibrium payoff for both players. Contributions in the intermediate
equilibrium increase as ° falls, while the highest and the lowest equilibrium contribution
remain unchanged. This explains also why one can get only weak monotonicity of x in
Proposition 4.2. For high uncertainty however, e.g., ° 0 , only the minimum contribution
equilibrium x¤¤¤(° 0 ) = 0 exists. The economic intuition for this result is clear. As uncertainty increases, ° declines, players fear that the decline in contributions of the other
players will erode the benefit of their own contributions. Thus, they will reduce their
contributions and eventually stop contributing at all. Note that this corresponds to the expost Pareto inferior equilibrium without uncertainty. This collapse in contributions may
come quite abruptly at some critical point.
6. Comparison with Other Approaches
In this section we compare our approach with two alternative ways of modelling uncertainty in games. Firstly we argue that Knightian uncertainty can have different effects to
risk aversion. Secondly we compare our approach with the technique of using a game of
incomplete information where an opponent can with some probability be a ‘‘crazy’’type.
23
Again we argue that our modelling approach can give different results and in addition
may be easier to interpret.
6.1
Risk Aversion
Example 6.1 considers a model of production externalities where individuals have constant degree of absolute risk aversion r.15 We show that changes in perceives uncertainty
(represented by decreases in °) have the opposite effect to increase in risk aversion.
Example 6.1 externalities
Consider two symmetric players who cooperate in producing a joint output z according
to the production function
z = f(x; y) := ®s ¢ (x + y)
where x and y denote the inputs of the two players, respectively, and ®s a random productivity parameter which takes the values
®h = 1 with probability ¼
and
®l = 0 with probability 1 ¡ ¼:
Without loss of generality, we assume x; y 2 [0; 1]: Players are assumed to be risk-averse
in regard to the uncertain joint output level z: Risk preferences of the players are given by
the CARA utility function
u(z) := 1 ¡ exp(¡r ¢ z):
The parameter r measures the average degree of risk aversion.
Finally, denote by c(x) the certain cost of input measured in terms of the output,
c(x) = k ¢ x:
Hence, one obtains the following symmetric (expected) payoff function where x is the action of the representative player and y the action of the opponent:
p(x; y) = ¼ ¢ [1 ¡ exp(¡r ¢ (x + y))] ¡ k ¢ x:
For players who are uncertainty-averse, the Choquet integral takes the following form:
P (x; y; °) = ° ¢ p(x; y) + (1 ¡ °)¢ min p(x; y)
y2[0;1]
= ¼ ¢ [1 ¡ exp(¡r ¢ x) ¢ [° ¢ exp(¡r ¢ y) + (1 ¡ °)]] ¡ k ¢ x;
with ° as uncertainty parameter: Optimising with respect to x yields the best-response
function
1
¼
½(y; °) = maxf0; ¢ [ln + ln[° ¢ exp(¡r ¢ y) + (1 ¡ °)]]g:
r
k
One can show that ½ is a decreasing and convex function of y: For ° = 0; it is a constant
function and for ° = 1 it is a linear function.
15
The example is the special case of the model in Cornes and Sandler (1996, p. 182-3) with
f (x) = k ¢ x, g(x) = (1 ¡ exp(¡r ¢ x)), and ®h = 1 and ®l = 0:
24
Figure 9 shows the best-reply functions for ° = 1; ° = 0:5 and ° = 0: In the left-hand
diagram the degree of risk aversion is r = 1 and in the right-hand diagram r = 2:
x
x
r=1
0
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
x¤1 x¤0:5
... ½(y; 0)
...
...
...
...
...
...
...
...
... ½(y; 0:5)
...
...
....½(y; 1)
...
y
0
x¤0
r=2
...
...
...
...
...
...
.
x¤1 x¤0:5
...
...
...
...
...
...
½(y; 0)
...
...
...
...
...
½(y; 0:5)
...
...
... ½(y; 1)
..
y
x¤0
Figure 9: Uncertainty aversion and risk aversion
This is a case of strategic substitutes with positive spillovers (Proposition 4.4). For both
degrees of risk aversion, r = 1 and r = 2; individual contributions x will rise in a
symmetric equilibrium under uncertainty as uncertainty increases and confidence falls.
On the other hand, for any fixed degree of confidence, ° = 1; ° = 0:5; or ° = 0, the
equilibrium contributions are smaller if there is more risk aversion r = 2; than in the case
of r = 1.
Example 6.1 has been chosen because it can show that more uncertainty may have the
opposite effect to risk aversion. The higher the uncertainty about the contribution of the
other player and, hence, about the output of the joint product, the higher the equilibrium
contributions will be. In contrast, the higher the risk aversion the lower contributions
will be.
Notice that uncertainty concerns the contribution of the other player, while it is the productivity of contributions that is risky. In both cases, a player faces uncertainty and risk
regarding the outcome of an input choice. Strategic uncertainty takes into account that the
players’behaviour may be incorrectly predicted. Hence, the worst case is a low contribution of the opponent which can be compensated by extra contributions of the player16 .
6.2
Irrational Types
It is possible to interpret equilibria under uncertainty as one state of a Bayes-Nash equilibrium with an irrational type of player (See, in particular, Mukerji and Shin (1999)).
16
In Eichberger and Kelsey (1999c), we have argued that uncertainty and risk aversion have a
different impact on free riding in the provision of public goods.
25
Consider, for example, a symmetric game and suppose that, with prior probability of
"; there is a type of player who chooses for each action of the other player the action
which minimizes the payoff of this player. One could model this by assuming a type of
player who has a payoff function which is the negative of the other player. The degree
of uncertainty ° corresponds to the prior probability of the irrational type ". With this
modification, an equilibrium under uncertainty in pure strategies for simple capacities
can be viewed as the play in the state of a Bayes-Nash equilibrium when both types are
rational.
We do not consider this as an equivalent model. In a Bayes-Nash equilibrium with irrational types, states of the world are combinations of types. Hence, with positive probability, there are states where ‘‘rational’’and ‘‘irrational’’types are matched. Complete
rationality of behaviour in a Bayes-Nash equilibrium is only possible if there are states of
the world when ‘‘rational’’players actually meet ‘‘irrational’’players. In an equilibrium
under uncertainty no such assumption is required. The fear of a bad outcome due to the
opponent’s behaviour is only in a player’s mind. Bayes-Nash equilibrium, if it wants to rationalise fears of players, has to assume an environment where these fears can come true.
The assumptions about bounded rationality implicit in the equilibrium under uncertainty
concept has to be replaced by assumptions about particular a priori states of the world.
In experiments or in observed behaviour, it may be difficult to distinguish these two approaches in particular if small probabilities are assumed. Situations with a high degree
of uncertainty correspond, however, to situations where ‘‘irrational’’types have a high
probability. If economic agents behave substantially more cautiously than predicted by
Nash equilibrium, then, in order to explain such behaviour by a Bayes-Nash equilibrium,
one has to assume a sufficiently high chance of ‘‘irrational types’’actually occurring.
A second reason why we consider an equilibrium under uncertainty as superior to a BayesNash equilibrium with irrational players, is the unjustified ad hoc assumption that each
player has an alter ego with completely antagonistic preferences. Our model of Knightian
uncertainty bases the cautious behaviour of players on assumptions about a decisionmaker’s preferences which, have been axiomatically derived and can, at least in principle,
be tested in experiments.
7. Concluding Remarks
We have argued that strategic uncertainty can be modelled by a Choquet integral of a
simple capacity. Uncertainty about the other players’ behaviour captured in this way
may help to understand why Nash equilibrium predictions are not always borne out in
experimental or in empirical studies.
Ideally, one would like to categorize the circumstances under which players will be more
or less uncertain about the their opponents’actions. One would expect there to be more
uncertainty in unfamiliar situations. Conversely there should be less uncertainty the more
frequently a similar decision has been made in the past. This would lead one to predict
that Nash equilibrium is a more appropriate tool for modelling interactions among anonymous individuals that take place frequently and customary. In contrast, behaviour in new
environments or in new institutions may be more appropriately modelled by equilibria
26
over uncertainty. Repeated interaction may thus not only help to make better predictions
but may also lead to more confidence about the actions of other individuals. This may
explain why institutions just as new products usually need some time before they can be
considered well established and reliable.
These comments raise however also several important questions. First of all, there is
no well-established rule of learning for capacities. It is not clear how one should adjust
beliefs that are modelled by capacities in the light of new information. How much new
information contributes to a better assessment of the situation and how much it contributes
to building up confidence remains an open question. There is no Bayes’law of updating
beliefs in the form of capacities.
A second question concerns the equilibrium concept. It is clear that the concept of a Nash
equilibrium, where beliefs coincide with actions is no longer applicable if beliefs are nonadditive. However it is far from clear what the appropriate constraints on consistency
between beliefs and actual behaviour should be. In this context, simple capacities may
be more than a convenient special case, since they allow us to distinguish the individual’s
probabilistic assessment of a situation, modelled by the additive part, from the degree of
uncertainty or confidence. In this case, uncertainty becomes a characteristic of a player
while the assessment of the situation is ref lected in the additive part of the simple capacity
for which one may require the traditionally strong notion of consistency incorporated in
Nash equilibrium.
The economic applications in this paper are not exhaustive. Careful evaluations of the
contributions that uncertainty can make to a better understanding of economic problems
need still be conducted. The intention was to demonstrate the potential of this approach to
produce new results that can be interpreted in a sensible way. Whether these new results,
based on the uncertainty of individuals about their social environment, provide a more
convincing and empirically verifiable explanation of economic phenomena remains to be
demonstrated. The conceptual framework however is available, and it is, in our opinion,
worthy of further investigation.
27
Appendix
This appendix contains the proofs of those results not already proved in the text.
Proposition 2.1. Choquet integral of a simple capacity
Consider a simple capacity º = ° ¢ ¼ +° ¢À where ¼ is an additive probability distribution
on a compact set X: The Choquet integral of a continuous function f on X has the
following form:
Z
Z
f dº := ° ¢ f d¼ + °¢ min f (x):
x2X
Proof. Let ¿ :=min f (x) be the smallest value of the function f on X: Since f is conx2X
tinuous and X is compact this minimum is well defined. By Definition 2.2,
1
R
R
f dº =
º(fx 2 Xj f(x) > tg) dt
0
+
R0
¡1
[ º(fx 2 Xj f(x) > tg) ¡ 1] dt:
Case 1: ¿ > 0 : Note that º(fx 2 Xj f(x) > tg) = 1 for all t < 0 and, hence,
Z0
[ º(fx 2 Xj f (x) > tg) ¡ 1] dt = 0
¡1
in this case. Furthermore, for any additive probability ¼ on a set X and any non-negative
real-valued function f on X; one has17
Z
Z1
f d¼ = ¼(fx 2 Xj f (x) > tg) dt
X
0
Hence, one obtains
1
R
R
R¿
f dº = º(fx 2 Xj f(x) > tg) dt+ º(fx 2 Xj f(x) > tg) dt
0
=
R¿
1 dt+
0
= ¿ + °¢
1
R
¿
1
R
¿
° ¢ ¼(fx 2 Xj f(x) > tg) dt
¼(fx 2 Xj f (x) > tg) dt
¿
1
R
R¿
= ¿ + ° ¢ [ ¼(fx 2 Xj f(x) > tg) dt¡ ¼(fx 2 Xj f (x) > tg) dt]
0
R0
R
= ¿ + ° ¢ [ f d¼ ¡ ¿ ] = °¢ f d¼ + (1 ¡ °) ¢ ¿:
X
X
Case 2: ¿ < 0 :
Note that, in this case,
g(x) := f(x) ¡ ¿ > 0;
17
For a proof see, e.g., Rudin (1987) pp. 172-3.
28
for all x 2 X: Hence, applying the argument of case 1 to g, one obtains
1
R
R
f dº = ¿ +
º(fx 2 Xj g(x) > tg) dt
0
= ¿ + °¢
1
R
R0
¼(fx 2 Xj g(x) > tg) dt
R
g d¼ = ¿ + ° ¢ [ f d¼ ¡ ¿ ]
= ¿ + °¢
R X
= °¢ f d¼ + (1 ¡ °) ¢ ¿ :
X
X
Proposition 3.2. If, for all players i 2 I;
(i) the strategy sets Si are compact and convex, and if
(ii) the payoff functions pi (si ; s¡i ) are continuous in s and quasi-concave in each player’s
own strategy si , then there exists an Equilibrium under Uncertainty in pure strategies.
Proof. Define the best reply correspondence of player i as
½i (s) :=arg max Pi (si ; s¡i ; ° i ):
si 2Si
Q mj
Since pi (si ; s¡i ) is a continuous function and since S¡i is a compact subset of
R ;
j6=i
continuity of
Pi (si ; s¡i ; ° i ) := ° i ¢ pi (si ; s¡i ) + (1 ¡ ° i )¢ min pi (si ; s¡i )
s¡i 2S¡i
follows from the maximum theorem (Berge (1963) ). From the maximum theorem, one
has also that ½i (s) is a well-defined, upper-hemi-continuous and compact-valued correspondence.
Viewing min pi (si ; s¡i ) as a set of functions parametrised by s¡i ; it is clear that
s¡i 2S¡i
min p(si ; s¡i ) is quasi-concave as the minimum of quasi-concave functions. Hence,
s¡i 2S¡i
Pi (si ; s¡i ; ° i ) is quasi-concave in si : Convex-valuedness of ½i (s) now follows from standard arguments.
Hence, ½ : S ! S defined by
½(s) := £ ½i (s)
i2I
is a compact and convex-valued, upper-hemi-continuous correspondence which has a
fixed point s¤ 2 ½(s¤) by the Kakutani fixed-point theorem. Clearly,
s¤i 2 ½i (s¤) =arg max Pi (si ; s¤¡i ; ° i )
si 2Si
is a Nash equilibrium of the game ¡(° 1 ; :::; ° I ) and, hence, an Equilibrium under Uncertainty in pure strategies.
Lemma 4.1. Let ° i > 0 for all i 2 I: Given Assumption 4.1, for all i; j 2 I; i 6= j and
all si 2 Si ; sj 2 Sj ;
29
1. if a game is characterised by positive (negative) spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0;
@si @sj
2. if a game has positive spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0;
@si @° i
3. if a game has negative spillovers, then
@ 2Pi (si ; s¡i ; ° i )
> 0:
@si @° i
Proof. By Assumption 4.1, strategy sets are intervals of R; Si = [ai ; bi ] for some real
numbers ai 6 bi : Hence, S = [a; b] where a = (a1 ; :::; aI ) 6 (b1; :::; bI ) = b: If the
game has positive spillovers, then fa¡i g =arg min pi (si ; s¡i ). For negative spillovers
fb¡i g =arg min pi (si ; s¡i ):
s¡i 2S¡i
s¡i 2S¡i
1. If a game has positive or negative spillovers, then there is a unique minimiser b
s¡i ;
fb
s¡i g =arg min pi (si ; s¡i ); which is independent of si : Differentiating Pi (si ; s¡i ; ° i )
s¡i 2S¡i
with respect to si ; one has
@pi (si ; s¡i )
@pi (si ; b
s¡i )
@Pi (si ; s¡i ; ° i )
= °i ¢
+ (1 ¡ ° i ) ¢
:
(A1)
@si
@si
@si
Differentiating with respect to the strategy of any player j 6= i; yields
@ 2 pi (si ; s¡i )
@ 2Pi (si ; s¡i ; ° i )
= °i ¢
>0
@si @sj
@si @sj
by Assumption 4.1.
2. For positive spillovers, b
s¡i = a¡i : Differentiating the first derivative (A1) with respect
to ° i yields
@pi (si ; s¡i ) @pi (si ; a¡i )
@Pi (si ; s¡i ; ° i )
=
¡
> 0;
@si @° i
@si
@si
2
i ;s¡i )
> 0 and s¡i > a¡i :
since @ p@si (si @s
j
3. For negative spillovers, recall ° i = 1 ¡ ° i : Hence, (A1) can be written as
@Pi (si ; s¡i ; ° i )
@pi (si ; s¡i )
@pi (si ; b¡i )
= (1 ¡ ° i ) ¢
+ °i ¢
:
@si
@si
@si
Differentiating with respect to ° i ; we have
@Pi (si ; s¡i ; ° i )
@pi (si ; b¡i ) @pi (si ; s¡i )
=
¡
> 0;
@si @° i
@si
@si
since
@ 2 pi (si ;s¡i )
@si @sj
> 0 and s¡i < b¡i :
Proposition 4.1. Let si 2 ½i (s¡i ; ° i ) and s0i 2 ½i (s¡i ; ° 0i ): Given Assumption 4.1,
30
(A1’)
1. if a game has positive spillovers, then
° i > ° 0i implies si > s0i ;
2. if a game has negative spillovers, then
° i > ° 0i implies si 6 s0i :
Proof. 18
By Assumption 4.1 and Lemma 4.1, the strategy sets Si are intervals of
R and the payoff functions Pi : S £ [0; 1] ! R are twice continuously differentiable
with positive cross derivatives for all variables. Hence, by Theorem 4 in Milgrom
and Roberts (1990), the game ¡(°) = (I; (Si ; Pi (¢; ° i ))i2I ) is supermodular. This
implies that all Si are lattices and all Pi are supermodular functions in all variables. By
supermodularity, for any two (si ; s¡i ; ° i ); (e
si ; s¡i ; e
° i ) 2 S £ [0; 1];
Pi (si ; s¡i ; ° i ) + Pi (e
si ; s¡i ; e
° i ) 6 Pi (maxfsi ; e
si g; s¡i ; maxf° i ; e
° i g)
+Pi (minfsi ; sei g; s¡i ; minf° i ; e
° i g):
Hence, for si > e
si and ° i 6 e
°i;
Pi (si ; s¡i ; e
° i ) ¡ Pi (e
si ; s¡i ; e
° i ) > Pi (si ; s¡i ; ° i ) ¡ Pi (e
si ; s¡i ; ° i );
i.e., Pi (si ; s¡i ; ° i ) ¡ Pi (s0i ; s¡i ; ° i ) is non-decreasing in ° i :
For the case of positive spillovers, the result follows now from Topkis’Monotonicity Theorem as stated in Milgrom and Roberts (1990). For the case of negative spillovers,
the same argument can be made in terms of ° i :
Proposition 4.2 Given Assumption 4.1,
1. there exists s¤ (°); s¤(°) 2 S ¤ (° 1; :::; ° I ) such that
s¤ (°) 6 s¤ (°) 6 s¤(°);
2. if a game has positive spillovers, then ° > ° 0 implies
s¤ (°) > s¤ (° 0 ) and s¤ (°) > s¤ (° 0 );
3. if a game has negative spillovers, then ° > ° 0 implies
s¤ (°) 6 s¤ (° 0 ) and s¤ (°) 6 s¤ (° 0 ):
Proof.
The proof of Proposition 4.1 shows that ¡(°) is a supermodular game. The result is
therefore a straightforward application of Theorem 5 and Theorem 6 in Milgrom and
Roberts (1990).
Lemma 4.2 For ¡ = (f1; 2g; (Si ; pi )i=1;2) ; let Se1 := S1 ; Se2 := ¡S2; pe1(e
s1 ; e
s2 ) :=
p1(e
s1 ; ¡e
s2); and pe2 (e
s1 ; e
s2) := p2(e
s1 ; ¡e
s2):
³
´
e
e
1. If strategies in ¡ are strategic substitutes, then strategies in ¡ = f1; 2g; (Si ; pei )i=1;2
18
The results on the monotonicity of best response correspondences and fixpoints in supermodular games,
which are used in this appendix, are based on the work by Topkis (1979) and Tarski (1955). These
results have been applied to economic and game theoretic models by Milgrom and Roberts (1990,
1994), Vives (1989) and others. All necessary concepts and results for our proofs can be found in
Milgrom and Roberts (1990).
31
are strategic complements.
2. Moreover, if ¡ has positive (negative ) spillovers then
² pe1 has negative (positive ) spillovers,
² pe2 has positive (negative ) spillovers.
Proof.
1. If strategies in ¡ are strategic substitutes then, by Definition 4.1,
for all si 2 Si ; sj 2 Sj ; j 6= i:
Differentiating pe1 with respect to e
s1 and e
s2 ; one has
2
@ 2 p1 (e
s1 ; ¡e
s2 )
@ pe1 (e
s1 ; e
s2)
=¡
> 0:
@e
s1 @e
s2
@s1 @s2
@ 2 pi (si ;s¡i )
@si @sj
< 0 holds
2
pe2 (e
s1 ;e
s2 )
Similarly, one argues for @ @e
> 0:
s1 @e
s2
2. Differentiating pe1 with respect to e
s2 and pe2 with respect to se1 ,
@ pe1 (e
s1; se2 )
@p1 (e
s1 ; ¡e
s2 )
@e
p2(e
s1 ; e
s2)
@p1 (e
s1; ¡e
s2 )
=¡
and
=
:
@e
s2
@s2
@e
s1
@s1
Proposition 4.3 Let si 2 ½i (sj ; ° i ) and s0i 2 ½i (sj ; ° 0i ); i 6= j; and suppose that strategies
in ¡ are strategic substitutes.
1. If a game has positive spillovers, then
° i > ° 0i implies si 6 s0i :
2. If a game has negative spillovers, then
° i > ° 0i implies si > s0i :
e be defined as in Lemma 4.2. For i = 1; 2; consider
Proof. Let ¡
Pei (e
s1 ; e
s2 ; ° i ) = ° i ¢ pei (e
s1; se2 ) + (1 ¡ ° i )¢ min pei (e
s1; se2 ):
By Lemma 4.2, for i = 1; 2;
@ 2 Pei (e
s1 ; e
s2 ; ° i )
sej
@ 2pei (e
s1; se2 )
> 0:
@e
s1 @e
s2
@e
s1 @e
s2
1. Suppose that ¡ has positive spillovers. By Lemma 4.2, pe1 has negative spillovers.
e (e
s1 ;e
s2 )
Hence, f¡ag = arg min pe1 (e
s1 ; se2 ) with ¡a > ¡b: Differentiating @ P1@e
with
s1
= °i ¢
se2
respect to ° 1; one has
@ 2 Pe1(e
s1 ; e
s2; ° 1 )
@e
p1(e
s1 ; e
s2) @ pe1 (e
s1 ; ¡a)
¡
< 0:
@e
s1@° 1
@e
s1
@e
s1
Similarly, by Lemma 4.2, pe2 has positive spillovers. Hence, fag = arg min pe2 (e
s1 ; se2 )
=
se1
32
with a 6 b: Differentiating
@ Pe2 (e
s1 ;e
s2 )
@e
s2
with respect to ° 2 ; one has
s2)
@ pe2 (e
s1; e
s2 ) @ pe2(a; e
@ 2Pe2 (e
s1; e
s2 ; ° 2)
¡
> 0:
=
@e
s2 @° 2
@e
s2
@e
s2
e is supermodular in (e
Hence, ¡
s1 ; e
s2 ; ° 1 ; ° 2 ). By Proposition 4.1, for e
s1 2 arg max Pe1(¢; se2 ; ° 1);
0
0
0
se1 2 arg max Pe1 (¢; se2 ; ° 1 ); e
s2 2 arg max Pe2 (e
s1 ; ¢; ° 2 ); and se2 2 arg max Pe2 (e
s1 ; ¢; ° 02 );
° 1 > ° 01 implies e
s1 > e
s01;
° 2 > ° 02 implies e
s2 > e
s02 :
since there are positive spillovers with respect to ° 1 and ° 2 :
e ° 1 = 1 ¡ ° 1 and se1 = s1;
By the definition of the game ¡;
° 01 > ° 1 implies s1 > s01 :
Moreover, se2 = ¡s2 and
° 2 > ° 02 implies s02 > s2 :
2. By analogous reasoning, in a game ¡ with negative spillovers,
° i > ° 0i implies si > s0i :
Proposition 4.4
Let x 2 ½(y; °) and x0 2 ½(y; ° 0 ): Suppose ° > ° 0 : If a game has
1. positive spillovers and strategic complements or
negative spillovers and strategic substitutes, then
x > x0 ;
2. positive spillovers and strategic substitutes or
negative spillovers and strategic complements, then
x 6 x0 :
Proof.
1. Consider the case of strategic complements,
@ 2 p(x;y)
@x@y
> 0: If there are positive (or
@ 2 P (x;y;°)
@x@y
negative) spillovers, then also
> 0: Hence, P (x; y; °) is supermodular. The
strategy set S = [a; b] is an interval of R: For positive spillovers, fag = arg min
y
p(x; y) and
@ 2 P (x; y; °)
@p(x; y) @p(x; a)
=
¡
> 0;
@x@°
@x
@x
for y > a: It follows from Topkis’Monotonicity Theorem (Milgrom and Roberts
(1990)) that ° > ° 0 implies x > x0 : The reasoning for negative spillovers is analogous
arguing for ° instead of °:
2 p(x;y)
2. Consider now the case of strategic substitutes, @ @x@y
< 0: Define pe(e
x; ye) := p(e
x; ¡e
y ):
@ 2 pe(e
x;e
y)
Thus,
> 0 and Pe(e
x; ye; °) := ° ¢ pe(e
x; ye) + (1 ¡ °)¢ min pe(e
x; ye) is super@x
e@ ye
ye2[¡b;¡a]
33
modular. For positive spillovers of p(x; y), pe(e
x; ye) has negative spillovers: Hence,
f¡ag = arg min pe(e
x; ye) and
ye2[¡b;¡a]
@ pe(e
x; ¡a) @ pe(e
x; ye)
@ 2Pe(e
x; ye; °)
=
¡
>0
@e
x@°
@e
x
@e
x
for ye < ¡a: Applying Topkis’Monotonicity Theorem again, ° > ° 0 implies x
e>x
e0 :
Since ° = 1 ¡ ° and x
e = ¡x, we can conclude that ° > ° 0 implies x 6 x0 : The
reasoning for negative spillovers is analogous arguing now for ° instead of °:
Proposition 4.5
If a symmetric game is characterised by strategic substitutes, then
there is a unique symmetric Equilibrium under Uncertainty.
Proof. Suppose there are two symmetric equilibria x; x0 2 S ¤ (°); x 6= x0 . Then x 2
½(x; °) and x0 2 ½(x0 ; °) must be true. Define Pe(e
x; ye; °) as in the second part of the
proof of Proposition 4.4. Pe(e
x; ye; °) is supermodular on S £ ¡S = [a; b] £ [¡b; ¡a]:
Consider x
e2e
½(e
y ; °) := arg max Pe(e
x; ye; °) and x
e0 2 e
½(e
y0 ; °): It follows from Topkis’
x
e2S
Monotonicity Theorem that ye > ye0 implies x
e>x
e0 : Since x
e = x and ye = ¡y holds, one
0
0
has y > y implies x > x ; a contradiction to the assumption of an equilibrium x = y
and x0 = y 0 :
34
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37
Volkswirtschaftliche Reihe/Economic Series
Prof. Dr. Hermann ALBECK
Nationalökonomie, insbesondere Wirtschaftsund Sozialpolitik
http://www.uni-sb.de/rewi/fb2/albeck/
Prof. Dr. Jürgen EICHBERGER
Nationalökonomie, insbesondere Wirtschaftstheorie
http://www.uni-sb.de/rewi/fb2/eichberger/
Prof. Dr. Ralph FRIEDMANN
Statistik und Ökonometrie
http://www.wiwi.uni-sb.de/friedmann/
Prof. Dr. Robert HOLZMANN
Nationalökonomie, insbesondere Internationale
Wirtschaftsbeziehungen
http://www.wiwi.uni-sb.de/lst/iwb/
http://www.wiwi.uni-sb.de/lst/iwb/
PD Dr. Udo BROLL
Prof. Dr. Christian KEUSCHNIGG
Nationalökonomie, insbesondere Finanzwissenschaft
http://www.wiwi.uni-sb.de/fiwi/
Prof. Dr. Dr. h.c. Rudolf RICHTER
Nationalökonomie, insbesondere Wirtschaftstheorie
http://www.uni-sb.de/rewi/fb2/eichberger/richter/home-eng.htm
Prof. Dr. Dieter SCHMIDTCHEN
Nationalökonomie, insbesondere Wirtschaftspolitik
http://www.uni-sb.de/rewi/fb2/schmidtchen/
Prof. Dr. Volker STEINMETZ
Statistik und Ökonometrie
http://www.uni-sb.de/rewi/fb2/steinmetz/
9901 Christian KEUSCHNIGG
Public Finance
9902 Jürgen EICHBERGER
David KELSEY
Economic Theory
9903 Jürgen EICHBERGER
David KELSEY
Economic Theory
9904 Christian KEUSCHNIGG
Wilhelm KOHLER
Public Finance
9905 Aymo BRUNETTI
Beatrice WEDER
International Economics
Rationalization and Specialization in Start-up
Investment
Non-Additive Beliefs and Strategic Equilibria
Education Signalling and Uncertainty
Eastern Enlargement of the EU:
How Much is it Worth for Austria?
More Open Economies Have Better Governments
9906 Ralph FRIEDMANN
Markus GLASER
Statistics and Econometrics
9907 Rudolf RICHTER
Economic Theory
9908 Rudolf RICHTER
Economic Theory
9909 Udo BROLL
Kit Pong WONG
International Economics
9910 Udo BROLL
Kit Pong WONG
International Economics
9911 Roland DEMMEL
Christian KEUSCHNIGG
Public Finance
9912 Jürgen EICHBERGER
David KELSEY
Economic Theory
9913 Jürgen EICHBERGER
Werner GÜTH
Wieland MÜLLER
Economic Theory
9914 Jürgen EICHBERGER
David KELSEY
Economic Theory
Effects Of The Order Of Entry On Market Share,
Trial Penetration, And Repeat Purchases: Empirical
Evidence Or Statistical Artefact ?
A Note on the Transformation of Economic Systems
European Monetary Union: Initial Situation, Alter
natives, Prospects - -in the Light of Modern Institutional Economics
Capital structure, and the firm under uncertainty
Hedging with Mismatched Currencies
Funded Pensions and Unemployment
Free Riders do not like Uncertainty
Dynamic Decision Structure and Risk Taking
Uncertainty and Strategic Interaction in Economics