Stable Conventions in Hawk-Dove Games with Many
Players
Daniel H. Wood∗
January, 2012
Abstract
This paper investigates the evolution of conventions in hawk-dove games between more than
two players when multiple players could share the same payoff-irrelevant label, such as “blue”
or “green”. Asymmetric conventions where one role is more aggressive develop; which convention is more likely depends on how many players in the contest share each label. Conventions closer to a pure strategy equilibrium of the game are stochastically stable. This logic
offers one reason for the emergence of informal property rights. In disputes over property, individuals naturally separate into two roles: the possessor, who is unique, and non-possessors,
who can be numerous. If the value of objects is low relative to the cost of conflict over them,
this asymmetry favors the development of informal property rights conventions.
JEL Classification: C73, D23
Keywords: n-player games, hawk-dove, anti-coordination, stochastic stability, evolution, convention, informal property rights
∗
Department of Economics, Clemson University. email: [email protected]. I would like to thank Jerry Green,
Michael Kremer, and Al Roth for their suggestions and support. I also thank Attila Ambrus, Drew Fudenberg, Chuck
Thomas, Patrick Warren, and seminar audiences at Harvard and Clemson for helpful questions and comments.
1
1
Introduction
The two-player hawk-dove game, shown in Figure 1, has a unique equilibrium if players are identical ex ante, in which hawk is played V /C fraction of the time. When the players have distinct
identities such as “row” and “column”, there are also pure strategy equilibria in which one role
plays hawk and the other role plays dove. Evolutionary dynamics generally predict that a population with asymmetries will universally adopt one of the pure strategy equilibria as the shared
convention (Selten 1980).1 If these asymmetries are “nominal” – such as colors or labels assigned
to each role but not affecting players’ payoff functions – then either convention is equally likely to
develop.
Hawk
Dove
Hawk
V −C V −C
,
2
2
V, 0
Dove
0, V
V V
,
2 2
Figure 1: Normal Form of the Hawk-Dove Game with Two Players. 0 < V < C.
In this paper I extend hawk-dove games to consider several players competing for an indivisible
resource.2 With several players, there can be payoff-irrelevant asymmetries between roles that are
not purely nominal: some roles may be more numerous than other roles. For example, a single
contest could be between two “green” players and three “blue” players.
Consider a population whose members participate continually in these conflicts, where indi1
“Convention” means a shared pure strategy Nash equilibrium a la Young (1993) if one exists. However, sometimes
a pure strategy Nash equilibria does not exist in the class of games I analyze and in that case “convention” refers to
whatever shared equilibrium is reached in the long run.
2
Hauert et al. (2006) develop an n-player version of the hawk-dove game. Their aim is to provide a general
classification of the evolutionary dynamics of social dilemmas with n players. They do not consider asymmetries
between players, which is the major focus of this paper. An older paper, Schaffer (1988), analyzes evolutionarily
stable strategies in hawk-dove games with more than two players in a finite population, but also assumes players are
undifferentiated.
2
vidual members follow role-contingent pure strategies. Members of the population slowly update
their strategies, usually by switching to a better pure strategy, but sometimes by switching to a
worse strategy. A strategy is stochastically stable if the entire population is likely to eventually
adopt it when mistakes in updating are rare but not unheard of. I show that if a pure strategy
Nash equilibrium exists in these games, it is the unique stochastically stable equilibrium. If such
an equilibrium does not exist, generally the mixed strategy equilibrium that is “closer” to a pure
strategy equilibrium – in the sense of more a homogenous set of strategy choices by the players –
is uniquely stable.3 In contrast to the two-player case, populations that repeatedly engage in multiplayer hawk-dove games will develop conventions that favor particular payoff-irrelevant labels.
Those conventions involve less numerous roles being more aggressive if V /C is low, and the opposite if V /C is high. Section 2 illustrates these results with the case of repeated contests between
three randomly chosen players who before each contest draw straws from a set of two short and
one long straw. Section 3 defines multiplayer hawk-dove games more generally. Section 4 defines
the evolutionary modeling framework, which is similar to Kandori et al. (1993). Section 5 partially
characterizes stochastically stable equilibria in multiplayer hawk-dove games where each player
has a role and some players share each role.
The selection of particular conventions based on payoff-irrelevant characteristics sheds light on
likely conventions in the two player game. When there are nominal asymmetries between players
that can serve as coordination devices, equilibria that ignore those asymmetries are not evolutionarily stable (Maynard Smith and Parker 1976; Selten 1980). However evolutionary stability
does not predict which convention will evolve in cases with payoff-irrelevant and sometimes even
payoff-relevant labels.4 “Paradoxical” equilibria are theoretically stable, such as animals abandon3
Existence of pure strategy equilibria, where all players in a role play the same strategy, is not guaranteed for
general V and C. If V < C, no pure strategy equilibrium exists in the blue / green example.
4
A large biology literature on evolutionary stability with different asymmetries exists, beginning with Maynard
Smith and Parker (1976) and Hammerstein (1981), who examined single possibly nominal asymmetries and multiple
3
ing their territory to intruding rivals or larger animals losing conventionally to smaller animals. If
the two player model is an abstraction from occasional conflicts with several participants, the logic
of conflicts with many players would rule out the owner-intruder sort of paradoxical equilibrium.5
This logic is especially relevant for economics, since human social conflicts often occur between more than two players. Many types of de facto property rights exist without legal protection.6 Section 6 applies the results of Section 5 to understanding informal property rights. I show
that even in the absence of legal protection or other advantage, ownership will be stable if the value
of possessing an object is low enough relative to expected cost of fighting over it. The exclusive
nature of the possession relationship favors the development of informal property rights.
2
Example: Drawing Straws
This section analyzes the evolution of conventions for a series of three-player hawk-dove contests
in each of which the players are given a nominal asymmetry as possible coordination device. It
illustrates the more general results of the paper.
Consider an ex ante homogenous population of players with access to a sequence of discrete
valuable prizes. Each round a randomly selected group of three players compete for that round’s
asymmetries, respectively. One branch of the literature continues the analysis of payoff-revelant asymmetries (for
example, Enquist and Leimar (1987) or Mesterton-Gibbons (1994)).
Another branch in this literature does not focus on payoff-relevant asymmetries but instead how a player’s behavior
in a particular hawk-dove contest affects his future role. These include Grafen (1987), which argues that coordination
on asymmetries which leaves players in a particular role permanently disadvantaged in reproduction would leave that
role with no real cost to fighting – what Grafen terms the “desperado” effect. Mesterton-Gibbons (1992) examines
which strategies are evolutionarily stable when a player’s success in one round leads to him being the owner in a future
round, and finds different sets of parameters where any convention might be an evolutionarily stable strategy. Finally,
Kokko et al. (2006) endogenizes the value of territories and cost of fighting and shows that coordination on ownership
is favored by these feedback loops.
5
Straightforward calculations show that the big-small sort of paradoxical equilibrium will not be stochastically
stable when size affects players’ payoffs, although this sort will be evolutionarily stable for small payoff-relevant
asymmetries.
6
Ellickson (1991) is a classic study.
4
prize. Each player draws a straw from a set of one long and two short straws, and these players
then play a hawk-dove game. Players may condition their action on which straw they have drawn.
The prize winner is determined through the players’ simultaneous choice of either hawk or
dove actions. If no player chooses hawk, one randomly wins the prize and receives V , while the
others receive 0. If one or more plays hawk, a hawk randomly wins the prize, receiving V , while
other hawks suffer a cost of fighting C, and any doves receive 0. Expected payoffs are given in
Figure 2. Note that unlike the two player hawk-dove game, where hawk is a dominant strategy
when V > C, in the three player game, hawk is only dominant when V > 2C. When the value of
the prize relative to the cost of losing a fight is high enough, the contest could have the “capacity”
for two pure-strategy hawks in equilibrium.
Hawk
Dove
Two hawks
V /3 − 2C/3
0
One hawk and one dove
V /2 − C/2
0
Two doves
V
V /3
Figure 2: Three Player Hawk-Dove Payoffs. Expected payoff from playing hawk or dove as a function of
the action choice of the other two players.
Players in the population follow pure strategies that they update slowly. After each round, a
randomly selected player from the population switches her strategy, with probability π, to the best
response to the population’s distribution of strategies, or, with probability , to a random other
strategy. 0 < π < 1.
When straws are not available, a population steady state exists where some of its members
always play hawk and other players always play dove. Straws serve as coordination devices. With
straws, there are three steady states which correspond with the Nash equilibria of the game:
• one in which all players who draw a long straw play hawk and all players who draw a short
straw play dove (if V < C), or all players who draw a long straw play hawk and some
5
players who draw a short straw play dove, while others play hawk (if C < V < 2C);
• one in which straws are ignored (which is the same as the steady state without straws); and
• one in which players who draw a long straw play dove and some players who draw a short
straw play hawk while others play dove (if V < C), or all players who draw a long straw
play dove and all players who draw a short straw play hawk (if C < V < 2C).
In other words, in two steady states, drawing a long straw makes players either more or less aggressive and drawing a short straw makes players the reverse, while in the third steady state players
ignore straws.
2.1
Dynamics With = 0
Ordinarily behavior will evolve in the direction of the players’ best responses to how the rest of the
population behaves. The system without mistakes eventually reaches a limit point corresponding
to one of the steady states.
Let hl be the probability that a player who has drawn the long straw plays hawk, and hs be
the probability that a player with the short straw plays hawk. A long straw holder faces two
hawks with probability h2s , one hawk with probability 2hs (1 − hs ), and no hawks with probability
(1 − hs )2 .7 Integrating the expected payoffs in Figure 2 over these possibilities, a long straw holder
is indifferent between hawk and dove if
2V − hs (V + 3C) + h2s C = 0.
7
(1)
Assume, as is customary, that sampling with replacement is a good approximation of the distribution of behavior
that players face from other members of the population.
6
El
S dove, L hawk
hl = 1
S dove, L hawk
hl = 1
l i.c.
All mix
All mix
s i.c.
S mix, L dove
S mix, L dove
Es
hs = 1
3(a): Phase Diagram. Dots indicate steady
states of unperturbed dynamic. Arrows indicate direction of behavioral change with unperturbed dynamic.
hs = 1
3(b): Basins of Attraction. The basin of
attraction for the equilibrium in which long
straw holders are doves is shaded with dots,
while the basin of attraction for the equilibrium in which long straw holders are hawks
is shaded with diagonal lines.
Figure 3: Evolution of Strategies in the Three-Player Hawk-Dove Game with V < C.
Likewise, each short straw holder is indifferent if
4V − (hl + hs)(V + 3C) + 2hl hs C = 0.
(2)
The probabilities hl and hs are symmetric in equation (2) because a short straw holder interacts
with one long straw holder and one other short straw holder.
The indifference curves defined by equations (1) and (2) divide the system into four regions
with different directions of movement. These dynamics give rise to three limit points that correspond to the three Nash equilibria of the game. Figure 3(a) shows the state space and the directional
movement of the population’s strategy choices due to players switching to better strategy choices
over time. The labeled dots mark the limit points.
7
2.2
Dynamics With > 0
When is positive but small, the system spends almost all of its time in limit points that are
more difficult to leave via the -probability errors than other limit points. This section argues
(using techniques that will be more fully specified in Section 4) that rare mistakes cause limit
points corresponding to pure strategy Nash equilibria to be more likely to occur than limit points
corresponding to Nash equilibria where at least one role mixes.
An equilibrium is more likely to be reached quickly and hence to be stable if the number of
errors necessary to leave that equilibrium’s basin of attraction from the unperturbed dynamic is
greater than the maximum number of errors necessary to enter its basin of attraction. The regions
delineated by the indifference curves in Figure 3(a) are not themselves the basins of attraction.
For the system to enter the basin of attraction of a different limit point, one role must change its
behavior so much that both roles move toward the new limit point. Figure 3(b) shades the basin of
attraction of the aggressive short straw holder equilibrium with dots and the basin of attraction of
the aggressive long straw holder equilibrium with diagonal lines.
The line segments marked El and El in Figure 3(b) are proportional to the mininum number of
errors necessary to leave the two limit points at the edge of the state space. Es is shorter than El
since the s-player indifference curve curve is concave and symmetric over the 45◦ line. The system
will spend most of its time at the limit point where long straw holders play hawk, since that limit
point is left less frequently than the other limit points.
In this example, when the value of winning the prize is low enough relative to the cost of
fighting, the equilibrium favoring long straw holders is a pure strategy equilibrium, and this causes
it to be the unique stochastically stable equilibrium. This holds true more generally, as will be
shown in Section 5. Limit points in mixed strategies are less stable than limit points in pure
strategies. In this example, a pure-strategy equilibrium exists, but even where no pure-strategy
8
limit point exists, how close each mixed strategy is to the corresponding pure strategy determines
which mixed-strategy limit point is stable. In general, only one equilibrium is stable.
3 n-Player Hawk-Dove Games
It is straightforward to generalize the payoffs of the hawk-dove game to an arbitrary number of
players. There is some good outcome from winning the game, with value V , and bad outcomes
when multiple players are aggressive, with cost C. In the classic formulation of the game the
outcome of an escalated contest is either injury or winning an object, and the winner does not
suffer injury.
Definition 1. An n > 2-player hawk-dove game is a simultaneous move game with n players,
n
P
each of whom can play an action ai ∈ A = {H, D}. Letting H =
1(ai = H) be the number
i=1
of H actions, payoffs for each player are
ui (ai = D; a−i ) =



V,
with probability (1/n), if H = 0


0,
otherwise
and
ui (ai = H; a−i ) =



V,
with probability (1/H)


−C,
otherwise
where 0 < V < (n − 1)C.
This paper analyzes n-player hawk-dove games with two information sets.
Definition 2. A role r ∈ {x, y} is a possible information set in the game with a fixed number of
players nr who share that role each round.
9
Assume without loss of generality throughout that x’s are more frequent: nx > ny . Let hr
denote the probability that a randomly drawn player assigned role r plays hawk.
A player of type i will face a number of hawks drawn from Hi+j = B(ni − 1; hi ) + B(nj ; hj )
where B(n; x) is the binomial distribution with n trials of probability of success x.8 Then
ni +nj −1
E(ui (H)|hi , hj ) =
X
Pr(Hi+j = x)
x=0
E(ui (D)|hi , hj ) =
Pr(Hi+j = 0)
V + xC
1+x
V
ni + nj
where the (V + xC)/(1 + x) terms are the expected values of playing hawk when facing a fixed
number x of other hawks.
Denote by H ∗ = dV /Ce the maximum number of players – ignoring roles – that could play
hawk in a pure strategy Nash equilibrium of the multiplayer hawk-dove game.9 For example, if
V /C = 9/4, then H ∗ = 3. H ∗ is approximately the capacity of the contest for supporting hawks.
I restrict attention to H ∗ < n, since for higher values the game becomes a n-player Prisoner’s
Dilemma.
With two roles there are three Nash equilibria of the n-player game that correspond to the
three equilibria of the two-player game: one equilibrium in which x’s are more aggressive, one
equilibrium in which y’s are more aggressive, and one equilibrium in which both types are equally
aggressive. Role r being “more aggressive” means (if H ∗ ≥ nr ) that everyone in role r plays
hawk, or (if H ∗ < nr ) that players in role r are more likely to play hawks than players in the other
role.
Theorem 1 characterizes the mixed strategy Nash equilibria of the game. However, since the
8
Here the binomial distribution serves as an approximation for the correct hypergeometric distribution. This is
standard in the literature.
9
Recall that dxe = min{i ∈ ZZ|i ≥ x} and bxc = max{i ∈ ZZ|i ≤ x}.
10
evolutionary population in my model consists of players with pure strategies the equilibria in the
evolutionary analysis are polymorphic equilibria equivalent to mixed-strategy equilibria.
Theorem 1. In n-player hawk-dove games with two roles x and y, there are three Nash equilibria:
i. hx = hy < 1;
ii. hy = 1 and hx < 1 (if H ∗ > ny ), or 0 < hy < 1 and hx = 0 (if H ∗ ≤ ny );
iii. hx = 1 and hy < 1 (if H ∗ > nx ), or 0 < hx < 1 and hy = 0 (if H ∗ ≤ nx ).
No other equilibria exist.
Proof. The equilibrium strategies for role r depend on the payoff difference between the two possible actions vr (hr , h−r ) ≡ E(ur (H) − ur (D)|hr , h−r ).
Clearly there exists an m such that vx (m, m) = vy (m, m). Now consider the asymmetric
case. If H ∗ > nr , then from the definition of H ∗ , there is an h−r where vr (1, h−r ) ≥ 0 and
v−r (1, h−r ) = 0. Likewise, if H ∗ < nr , there is an hr such that vr (hr , 0) = 0 and v−r (hr , 0) < 0.
Finally, consider hx 6= hy and assume towards contradiction that hx , hy ∈ (0, 1) and vx (hx , hy ) =
vy (hy , hx ) = 0. An x-player faces a different distribution of hawks than a y-player, so it cannot be
the case that vx (hx , hy ) = vy (hy , hx ). Thus no other equilibria exist.
Translated into polymorphic equilibria in pure strategies, for low enough H ∗ , every asymmetric
equilibrium involves some players of both types playing dove. For high enough H ∗ , in every
asymmetric equilibrium some players of both types sometimes play hawk. For intermediate H ∗ ,
the equilibrium where y’s are more aggressive involves some x’s playing hawk, but the equilibrium
where x’s are more aggressive involves no y’s playing hawk.
11
4
Evolution and Stable Equilibria
This section describes the evolutionary dynamic of the model. There is a population of K players,
where K is large. Each round n players are drawn from the population and assigned roles. The n
players then play a hawk-dove game and one player updates his strategy for the next time he will
play. The state space of the system, Θ, consists of the number of players playing each strategy.
A strategy S is a choice of action from A = {H, D}, contingent on the payoff-irrelevant group
membership of the player, g ∈ G = {X, Y }. S : G → A, so there are four possible strategies and
Θ is a finite subset of N3 . The state can also be represented as a point in [0, 1] × [0, 1], where the
x-dimension is hx and the y-dimension is hy .
Each period one set of players is drawn with replacement from the population and plays a
hawk-dove game. At the end of the period, one agent then reevaluates her strategy. Every player
is equally likely to be drawn for the contest and then again every player is equally likely to be
selected to reevaluate her strategy. Different sampling assumptions could produce different results,
as in Robson and Vega-Redondo (1996). Let si, t denote player i’s strategy at time t and δt be
a vector expressing the empirical distribution of strategies in the population. Let s∗ (δ) be the
best response to a distribution of strategies δ and s∗ (δ) a random non-best response to δ.10 When
player i chooses a new strategy, if si, t 6= s∗ (δt ), then with probability π ∈ (0, 1) she switches to
s∗ (δ). This is the unperturbed evolutionary dynamic. The perturbed dynamic introduces a second
possibility, namely that with probability , when i updates she mistakenly switches to a strategy
Generically s∗ (δ) will be unique. Players will be indifferent between hawk and dove at a particular probability of
facing a hawk. Since all of the possible probabilities are ratios between the number of opponents playing hawk and
the number of possible opponents, best responses are unique for all but measure zero of parameters.
10
12
that is not a best response to δt .
si, t+1




si, t




= s∗ (δ−i, t )






s∗ (δ−i, t )
with probability 1 − π − with probability π
.
(3)
with probability Fixing π, let P be the Markov transition matrix on Θ associated with the probabilistic bestresponse dynamic when = 0, and let P be the Markov transition matrix on Θ for a particular
> 0. Each P gives a new probability distribution over states µt+1 = P µt as a function of
a current period distribution µt . The perturbed Markov process when > 0 is ergodic, and so
µ = P µ exists and is unique. This distribution of probabilities is reached almost surely from any
initial state µo if enough time passes. Stochastic stability focuses on the distribution µ as errors
become rare:
lim µ = µ∗ .
→0
From Young (1998, Theorem 3.1), µ∗ exists. Stochastically stable states s are those with strictly
positive probability in the limit distribution µ∗ : µ∗ (s) > 0.
Stochastically stable states are limit points of the unperturbed process P . Ellison (2000, Theorem 1) provides a convenient sufficient condition for the stochastically stable strategy to be a
particular limit point, though comparison of the basins of attractions of every limit set. An equilibrium is stable if the minimum number of errors necessary to leave that equilibrium’s basin of
attraction (the basin of attraction’s radius) is greater than the maximum number of errors necessary
to enter its basin of attraction (the basin of attraction’s coradius).
The formal definition of the radius and coradius of a point in the state space can be given using
the cost function C(s, s0 ), the minimum number of probability events required to move from a
13
point s to a point s0 in the state space. Let ω designate a point in a limit set of the unperterbed
process. Let the set B(ω) be all points in the basin of attraction of a limit point ω.
R(ω) = 0min C(ω, s0 )
CR(ω) = max
C(s0 , ω)
0
s ∈Θ
s 6∈B(ω)
Then if R(ω) > CR(ω), the point ω is stochastically stable.
5
Stability of Pure versus Mixed Strategy Equilibria
This section demonstrates – analytically when H ∗ is low and through numerical methods when it
is not – that when there are two roles and n ≥ 3, equilibria “close enough” to pure strategy Nash
equilibria are stochastically stable. It builds on intermediate results showing that the region where
hawk is a best response is convex in other players’ mixed strategies (Lemma 1) and that totally
symmetric equilibria are not stable (Lemma 2).
Lemma 1. If s is a vector of probabilities that each other player will play hawk and s0 is another
vector of such probabilities, where for α ∈ (0, 1) and i 6= j,
s0i = αsi + (1 − α)sj
and
s0j = αsj + (1 − α)si .
then
i) if H is a best response to s, then H is the best response to s0 .
ii) if D is a best response to s0 , then D is the best response to s.
Proof. Let v(x) denote the difference in expected payoff from playing H rather than D when there
are x other players who play H. v(x) ≡ (V −xC)/(x+1) is strictly decreasing and strictly convex
14
in x.11
Let F be the cumulative distribution function for the number of hawks induced by s. F 0 ,
the cumulative distribution function for the number of hawks induced by s0 , is a mean-preserving
spread of F , and so F second-order stochastically dominates F 0 . The strictly convex function v
has higher expectation under F 0 than under F , so
Z
Z
0
v(x)dF (x)dx >
v(x)dF (x)dx.
If a player’s best response to s is hawk, then
Z
v(x)dF (x)dx ≥ 0
and so it follows that the player’s best response to s0 is also hawk. Alternatively, if
Z
v(x)dF 0 (x)dx = 0,
then
Z
v(x)dF (x)dx < 0,
and so the player’s best response to s is dove.
Lemma 1 captures the reason why Es is shorter than El in Figure 3(b). E(ui (H)−ui (D)|hi , hj )
is quasiconcave in hi and hj , so that for every role, the region where hawk is the best response is
11
v(0) 6= V , because then the expected payoff to D is greater than 0. However,
V
V +C
V
V +C
V −C
− V −
=−
+
< v(2) − v(1) = −
v(1) − v(0) =
2
n
2
n
6
when n ≥ 3, so v(x) is convex on [0, ∞).
15
convex. Lemma 1 can be applied more conveniently through lines in (hx , hy )-space that hold
constant the expected number of hawks a player faces. If a player is indifferent to n players all
playing hawk with probability x, he prefers playing dove to any other distribution where m < n
players play hawk with probability xn/m.
Lemma 2. A limit point where the probability of playing hawk is independent of a player’s role is
not stochastically stable in large populations.
Proof. Let v(m) = E[ui (H) − ui (D)|m, m] where m is the identical probability that n − 1 other
players play hawk. There is an m∗ such that v(m∗ ) = 0. Let KH = (K − 1)m∗ is the number
of players in a population of size K playing hawk that would produce hawk with probability m.
For almost all parameter values, KH is not an integer. Because of the definition of m∗ , the best
response to n − 1 strategies selected at random from a population of size K with dKH e hawks is
dove, and the best response to n − 1 strategies selected at random from a population of size K with
bKH c hawks is hawk.
Denote the number of players who play hawk when x and dove when y by KHD , and so on.
In the totally mixed equilibrium, both roles face the same distribution of hawk actions: dKH e =
KHH + KHD = KHH + KDH . The basin of attraction of a limit point ωM corresponding to
the totally mixed equilibrium does not include the state where KHH + KHD = dKH e + 1 and
KHH + KDH = dKH e − 1 because that point is in the basin of attraction of ωX . From ωM it
takes at most two errors to move to this point outside ωM ’s basin of attraction. The other basins of
attraction have radii proportional in size to K, so in large populations R(ωM ) is less than R(ωX )
and R(ωY ).
Totally mixed limit points are saddle points of the dynamic system with = 0. Small movements in an appropriate direction take the system out of these limit points’ basins of attraction.
16
Eyd
ωy
hy
y i.c.
Eyh
ωm
Exd
x i.c.
ωx
hx
Exh
Figure 4: Theorem 2 Notation.
When H ∗ , the number of players playing hawk in a pure-strategy equilibrium of the game is
low, the asymmetric equilibrium in which y’s are more aggressive is closer to the pure-strategy
equilibrium. Both asymmetric equilibria involve the more aggressive role mixing between hawk
and dove, and if y’s are the more aggressive role, more of them play hawk than x’s would if x’s
were the more aggressive role.
Theorem 2. In n-player hawk-dove games with nx > ny , if H ∗ ≤ ny the unique stochastically
stable equilibrium is for all players with role y to play hawk and all players with role x to play
dove.
Proof. Let ωr be the limit point with role r playing hawk with higher probability than the other
role, and ωm be the limit point where both roles play hawk with the same probability. Let Exh
denote the distance necessary to leave ωx by players switching from a strategy of hawk when x
to dove when x, and Exd denote the distance necessary to leave ωx by players switching from
a strategy of dove when y to hawk when y. Likewise let Eyh denote the distance necessary to
17
leave ωy by players switching from a strategy of hawk when y to dove when y, and Eyd denote
the distance necessary to leave ωy by players switching from a strategy of dove when x to hawk
when x. R(ωx ) is proportional to min(Exh , Exd ) and R(ωy ) is proportional to min(Eyh , Eyd ). See
Figure 4.
First, nx > ny implies Exd > Exh : at ωm an x-player faces n − 1 other players playing
hawk with probability m. Consider a mean-preserving contraction of the distribution of hawks at
ωm with nx − 1 hawks playing hawk with probability m(n − 1)/(nx − 1). x-players facing this
distribution will play dove, and Exh is the distance in the x dimension between ωx (where x-players
are indifferent between hawk and dove) and ωm , so
Exh
m(n − 1)
<
−m=m
nx − 1
ny
nx − 1
≤ m = Exd
and so R(ωx ) ∝ Exh .
Second, Exd = Eyd = m: Exd is the distance between ωx and ωm in the y dimension, while
Eyd is the distance between ωy and ωm in the x dimension. ωm is on the 45◦ line, so these distances
are the same.
Third, nx > ny also implies Eyh > Exh : at ωy , ny players play hawk with probability Eyh + m,
and at ωx , nx players play hawk with probability Exh + m. Because nx > ny , Eyh + m > Exh + m.
Hence, Eyd = Exd > Exh and Eyh > Exh , so R(ωy ) = min(Eyh , Eyd ) > R(ωx ) = Exh .
The stochastically stable equilibria are the ones closer to the pure strategy equilibrium of the
n-player game (which might not exist given the particular number of players in each role). In the
low-value cases that Theorem 2 characterizes, ny is always closer to the capacity H ∗ than nx is.
With higher ratios of V /C, which asymmetric equilibrium is closer to a pure strategy equilibrium still predicts which equilibrium will be stable. It is difficult to find closed-form solutions
18
R
R(ωy )
R(ωx )
0
ny
nx − 1
n V /C
Figure 5: Relationship Between R(ωy ) and R(ωy ) and Number of Players in Each Role.
for equilibria in n-player hawk-dove games, though for H ∗ low enough, the stochastically stable
equilibria can be found based on the properties of the indifference curves. This technique does not
work for higher stakes. Instead I solve numerically for the limit points of the system and calculate
the implied radii.12
These numerical calculations show that R(ωy ) achieves its maximum when V /C = ny and
R(ωx ) achieves its maximum when V /C = nx − 1, as shown in Figure 5. For each role, R(ωr )
is increasing until the maximum and then decreasing thereafter. The two radii are equal at at
V /C ≈ (nx + ny + 1)/2, the average of the points at which the two maximas occur. The intuition
for the relationship between V /C and R(ωy ) is that at V /C = ny , ωy = (0, 1), after which ωy
starts increasing in the x-dimension at a more rapid rate than ωm is increasing, which causes Eyd
to fall. The intuition with R(ωx ) is similar. The point V /C = nx − 1 is the lowest ratio at which
ωx = (1, 0). Further small increases in V /C only shift ωm , causing Exh to fall.13
Thus, if a pure-strategy equilibrium exists, it will be stochastically stable. For parameters
12
I choose nx and ny pairs from a grid n ∈ {1, 2, 3, 4, 6, 8, 10, 15, 20, 30, 50}. I take every combination of
nx ∈ n and ny ∈ {n|n < nx }. For each combination I then iterate over 200 equally spaced values of V /C in the
interval [0, nx + ny − 1].
13
Note that when nx = ny + 1, R(ωx ) and R(ωy ) achieve their maxima at the same V /C ratio. In fact, in this case,
for V /C where H ∗ > ny , R(ωx ) = R(ωy ). As Section 2’s example, the x-player indifference curve is symmetric
across the 45◦ line.
19
close enough to these cases, whichever role is closer in number of players to H ∗ is more stable.
Furthermore, when nx 6= ny + 1, there is a unique stable equilibrium. It follows that if ny , V , and
C are fixed, there is some number of x players such that the equilibrium favoring the less numerous
y players will be stable.
6
Ownership Norms, Paradoxical Strategies, and Coordination
When modeled as a two-player game, contests over possessions or territory are equally likely to
evolve a convention that disadvantages the present owner of the object as a convention that favors
her. If contests sometimes involve more than two players, the previous stability results provide a
reason that informal property rights over rivalrous objects would exist. Objects’ possessors can
be sure that they are the unique possessor. Non-possessors have a relationship to a given object is
shared with all other non-possessors.
Define an object o as a vector of the benefits of using it, V , and the intensity of fights over it,
which produce cost C. The possessor of an object o at period t in a repeated hawk-dove game is
the player who received utility V from using the object o at the end of period t − 1. The possessor
of the object knows that she is the possessor and that no one else is the possessor. Possession is
conceptually distinct from ownership: the owner of an object o in a repeated hawk-dove game is
a player who possesses the object at period t and will possess it with greater likelihood than any
other player in period t + 1. Ownership norms are equilibria in which objects have owners.
In an economy formed by K people repeatedly playing n-player hawk-dove games to determine possession of each of a set of objects, where peoples’ behavior evolves as specified in Section
4 and where people condition their actions on whether they were a possessor or not in the prior
20
period, for objects with high enough costs of fighting or sufficiently many non-possessors relative to possessors, ownership norms are stable. If V /C < 1, there will be a pure-strategy Nash
equilibrium where the one possessor plays hawk and all non-possessors play dove. That equilibrium is the unique stochastically stable equilibrium. The numerical results in Section 5 show that
ownership norms are uniquely stable in cases of slighly higher value, where the possessor-favoring
equilibrium also involves non-possessors contesting for an object, but less aggressively.
That increasing the cost of fighting increases the likelihood of ownership norms evolving is
not particularly surprising, as it increases the cost of coordination failure. The comparative static
on the number of possessors is more surprising, but shares a similar logic. A convention where
possessors give up an object is worse the more non-possessors there are for the object because
of greater coordination failure when many non-possessors play hawk.. This renders paradoxical
strategies where possession reduces the chance of future ownership less stable.
Of course, possessing an object can change the possessor’s payoffs in future contests over
it. If people are better at defending their possessions than taking other peoples’ possessions, for
instance, then this asymmetry in expected payoffs also favors informal property rights. I do not
intend to claim that the possession relationship is the predominant source of ownership norms, but
rather that it favors development of these norms even when other forces did not favor them. Thus,
analysis of norms in the multiplayer hawk-dove game yields a new conceptual understanding of
possession and ownership.
There is considerable evidence that possession is used as a coordination device. Original possession is the origin of property in common law in cases such as wild animals, sunken ships, or
fugitive resources. Rose (1985) argues that this doctrine developed because possession is an especially clear form of communication that something is owned. In addition, many of the shortcomings of extant informal property rights come from the limitations of signalling ownership through
21
possession. Field (2007) finds that the most economically significant impact of a titling program
in Peru was that it allowed new formal owners to shift from working at home to working in the
market. Likewise Goldstein and Udry (2008) find that farmers in Ghana with less secure property
rights (due to kinship structure) leave land fallow for inefficiently short periods of time in order to
more securely maintain informal rights to farm it.
An interesting but speculative question to ask is if these results have any connection to why
norms about intellectual property are weaker in many segments of the U.S. population than norms
against physical theft. The rivalrous nature of physical property can encourage stronger ownership norms as a way of avoiding conflict and so respect of physical possessions arises naturally,
while intellectual property, being non-rivalrous, does not require coordination among its users.
Intellectual property must place greater reliance on legal enforcement for its strength.
7
Conclusion
I study the conventions that evolve in the long-run in hawk-dove games when players have different
roles that vary in frequency but do not affect payoffs. When the evolutionary process is modeled
as involving continous small probabilities of non-optimal changes in behavior a la Kandori et al.
(1993), conventions where one role sometimes plays hawk and sometimes plays dove usually are
less stable than conventions closer to pure strategies. That differences in the frequency of roles
affects which conventions develop has not previously been remarked on in the large literature on
conventions in coordination games.
These stability results imply that the exclusive nature of possessions can stabilize an ownership
convention relative to alternative conventions where owners surrender ownership. One person
possesses a physical object at any point in time but possibly several people may wish to possess the
object in the future. When the conflicts determining future possession are modeled as one conflict
22
between many people, the only stable equilibrium, if objects are not too valuable, occurs when
current possessors maintain possession of their objects. Ownership norms develop even when
possession confers no inherent advantage in ability or motivation on the possessor in subsequent
contests.
23
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