History as a Coordination Device∗

Rossella Argenziano†and Itzhak Gilboa‡

October 2010

Abstract
Coordination games often have multiple equilibria. The selection of
equilibrium raises the question of belief formation: how do players gen-
erate beliefs about the behavior of other players? This paper takes the
view that the answer lies in history, that is, in the outcomes of similar
coordination games played in the past, possibly by other players. We
analyze a simple model in which a large population plays a game that
exhibits strategic complementarities. We assume a dynamic process
that faces different populations with such games for randomly selected
values of a parameter. We introduce a belief formation process that
takes into account the history of similar games played in the past, not
necessarily by the same population. We show that when history serves
as a coordination device, the limit behavior depends on the way history
unfolds, and cannot be determined from a-priori considerations.

1 Introduction
Games with strategic complementarities typically exhibit multiple equilib-
ria. The game theoretic literature has witnessed many attempts to select
equilibria based on the parameters of the game. The equilibrium selection
literature includes many notions that are defined by the game itself (see van
∗
We are grateful to Dirk Bergemann, Don Brown, Dino Gerardi, Stephen Morris, Barry
O’Neill, Ady Pauzner, Alessandro Pavan, Colin Stewart and Nicolas Vieille for comments
and references. Gilboa gratefully acknowledges support from the Israel Science Foundation
(Grant No. 975/03). This paper is part of the Polarization and Conflict Project CIT-
2-CT-2004-506084 funded by the European Commission-DG Research Sixth Framework
Programme.
†
Department of Economics, University of Essex. rargenz@essex.ac.uk
‡
HEC, Paris, and Tel-Aviv University. tzachigilboa@gmail.com

1
 2

Damme (1983)), such as the risk-dominance criterion. Other types of consid-

erations attempted to embed the game in a dynamic process (Young (1993),
Kandori, Mailath, and Rob (1993), Burdzy, Frankel and Pauzner (2001)) or
in incomplete information set-up (Carlsson and van Damme (1994)).
It is noteworthy that risk dominance has emerged as the preferred selec-
tion criterion based on quite different types of considerations. On the other
hand, the literature on strategic complementarities arising from network
externalities tends to favor Pareto dominant equilibria over risk dominant
ones (see Katz and Shapiro (1986)). This suggests a more agnostic view,
according to which the parameters of the game cannot, in general, predict
equilibrium selection. It appears that game theoretic considerations could
be used to impose certain restrictions on the possible outcomes, but the ac-
tual selection of an equilibrium is often left to history, chance, institutional
details, or other unmodeled factors.
In this paper we are interested in a dynamic process, according to which
large populations are called upon to play a simple coordination game. In the
first stage, each player chooses either a low or a high action. In the second
stage, nature chooses a low or a high outcome, and nature’s move depends on
the set of players choosing the high action. Consider the decision of a single
player in this game. The optimal action to take depends on his assessment of
the probability of a high outcome. We maintain that this assessment would
and should be based on the results of past instances of similar games. These
games may have been played by the same population or by others. Each past
game might differ from the current one by one parameter at most, which is
a proxy for the difference in the expected payoff of the two strategies. Both
the nature of the game and the identity of the population playing it should
be taken into account in the evaluation of the similarity of past games to the
present one. But ignoring these past games would hardly seem a rational
way of generating beliefs.1
1
The belief formation process may be embedded in a meta-game, which will also have
 3

Consider for example the following “revolution” game played by a large

population. In stage 1, each player i ∈ [0, 1] chooses whether to participate
in a revolutionary attempt, or to opt out. In stage 2, nature chooses a
move in {F, S}, which stand for F ailure and for Success of the revolution,
respectively. Nature’s move depends on the set of players who have chosen
to participate. After each player determined her choice of participation and
nature determined the success of the revolution, the game is over. The
payoff of each player depends only on her own choice of participation, and
on nature’s move. The payoff function u = ui for every i ∈ [0, 1] is given by
the following matrix:

S(uccess) F (ailure)
P articipate 1 0 (1)
x+1
Opt out 2 x

where x ∈ [0, 1] is the parameter of the game.

The interpretation of this matrix is as follows. The worst thing that can
happen to an individual in this game is to participate in a failed coup. The
result is likely to involve imprisonment, exile, decapitation, and the like.
This worst payoff is normalized to 0. The best thing that can happen to an
individual is that she participates in a revolution that succeeds. In this case
she is a part of a (presumably) better and more just society. This payoff is
normalized to 1.
An individual who decides to participate in the revolution therefore de-
cides to bet on its success with the extreme payoffs of 0 and 1. Between
these extreme payoffs lie the payoffs for an individual who decides to opt
out, foregoing the chance of being part of the revolution. The payoff of such
an individual still depends on the outcome of the revolutionary attempt.
Should this attempt fail, such an individual would get x, which is a measure
of the well-being of the people in the status quo. If, however, the revolution
a flavor of a coordination game. We assume, however, that people have a fundamental
tendency to expect the future to be similar to the past. To quote Hume (1748), "From
similar causes we expect similar effects."
 4

succeeds, even the individuals who were passive will benefit from the new
regime. However, not being part of the revolutionary forces, they would not
reap the benefits of revolution in its entirety. Their payoff would only equal
the arithmetic average between the full benefit, 1, and the status quo, x.
Consider the decision problem of a potential rebel. Imagine that rumors
have been spreading that the revolution would start tonight. She can ignore
the rumors and go to sleep, or take to the streets. For simplicity, assume
that this is a one-shot, binary decision. The potential rebel sits at home and
attempts to assess the probability that the revolution would succeed. How
would she do that? Suppose that it is common knowledge in the population
that revolution games of the type above have been played in the past. We
believe that the history of such similar games played in the past should affect
the beliefs of the potential rebel.
In this paper, we present a simple belief formation process for a class
of games that includes the “revolution” example above. The process is
such that the probability assigned to a high outcome in the current game
is the weighted empirical frequency of high outcomes in past games, where
the weights are given by a similarity function that takes into account the
differences between past games and the current one. We find that beliefs
that are history-dependent may lead to different behavior, depending on the
way history unfolds.

The rest of this paper is organized as follows. We first discuss related

literature. Section 2 describes the stage game. We devote Section 3 to
modeling the way players generate beliefs given history. Section 4 describes
the dynamic process and provides the main result of the paper. Finally,
Section 5 concludes.

1.1 Related literature

Our paper is closely related to the equilibrium selection literature discussed

in the introduction. It also relates to the literature on coordination games
 5

of regime change. The conceptualization of a revolution as a coordination

game dates back to Schelling (1960) at the latest. There exist alternative
conceptualizations in the political science literature, such as Muller and Opp
(1986), who emphasize the public good aspect of a revolution. Yet, the co-
ordination game model of a revolution has been the subject of many studies.
Lohmann (1994) studied the weekly demonstrations in Leipzig and the evo-
lution of beliefs along the process. More recently, Edmond (2008) studied
information manipulation in games of regime change, whereas Angeletos,
Hellwig, and Pavan (2007) focus on a learning process by which individuals
playing such games form beliefs. As in Lohmann (1994) and Angeletos, Hell-
wig, and Pavan (2007), we study the evolution of beliefs in a game that is
played repeatedly. However, as opposed to these papers, our game is played
by a new population at every stage. Thus, our focus is on the generation
of prior beliefs (over other players’ actions), based on similar games, rather
than on the update of already existing prior beliefs by Bayes’s law. Closely
related to the belief formation process that we study is the process studied
by LiCalzi (1995), which looks at the case where players give the same sim-
ilarity weight to the outcome of all the games in a given class. Jehiel (2005)
introduces a solution concept in which players form beliefs about their oppo-
nents’ behavior by grouping nodes in which the opponents play into analogy
classes. Finally, Steiner and Stewart (2008) study similarity-based learning
in games and show that contagion can lead to unique long-run outcomes.

2 The Stage Game

We describe a symmetric two-stage extensive form game Gx depending on a
parameter x ∈ X ≡ {x1 , ..., xJ } for J > 2. The cardinal values of the para-
meter x will be of no import, but their order will. There is a continuum of
players [0, 1]. In stage 1 all players move simultaneously. The set of moves
for each player i is Si = {0, 1}. In stage 2, after each player determined
her move in {0, 1}, nature chooses an outcome ν ∈ {0, 1}. Nature’s move
 6

depends on the set of players choosing 1 in stage 1, A ⊂ [0, 1]. Specifically, if

A is Lebesgue-measurable, we assume that nature chooses ν = 1 with prob-
ability ϕ(λ(A)) where ϕ is strictly increasing, with ϕ(0) = 0 and ϕ(1) = 1,
and λ stands for Lebesgue’s measure. If A is non-measurable, the probabil-
ity of nature choosing ν = 1 can be defined arbitrarily. At equilibrium, the
set A will be measurable.
After each player determined her choice and nature determined the out-
come (by the probability ϕ(λ(A))), the game is over. The payoff of each
player depends only on her own choice and on nature’s move. However, ex-
ante, the game exhibits strategic complementarities: the expected payoff of
a strategy si ∈ {0, 1} is strictly increasing in the measure of players taking
this strategy
Assume, then, that an individual i attempts to estimate the expected
utility of playing 1 versus 0 for a given game x ≤ xJ . Suppose that in-
dividual i’s belief over the measure of other individuals who choose 1 is
given by a measure μi,x over (the Lebesgue σ-algebra on) [0, 1]. That is, for
every Lebesgue-measurable set B ⊂ [0, 1], individual i assigns probability
μi,x (B) to the event that the measure of individuals who eventually choose
1 (with or without herself) lies in B. Specifically, the subjective probability
of individual i that nature will choose 1 in the game x is
Z
pbi,x = ϕ(p)dμi,x (p).
[0,1]

We assume that for every x ∈ X is there exists a unique p̄x ∈ [0, 1] such
that playing 1 is optimal if and only if player i believes that nature will
choose 1 with probability larger or equal to p̄x .
Given beliefs μi,x , player i’s expected payoff from playing 1 in game Gx
is greater (smaller) than her expected payoff from playing 0 iff pbi,x > p̄x
pi,x < p̄x ). For simplicity we assume that in case of a tie, pbi,x = p̄x , player
(b
i will play 0.2
2
While this assumption will prove immaterial, it simplifies analysis because a random
 7

We assume that p̄x is strictly increasing in x ∈ X. That is, the games

are assumed to be ordered according to the difference in the expected payoff
of the two strategies. We further assume that p̄x1 = 0 and p̄xJ = 1. That
is, in the game Gx1 , strategy 1 is dominant, whereas in GxJ — strategy 0 is.
At equilibrium, all players will have the same beliefs, hence pbi,x = pbx ,
i.e., it is independent of i. Therefore, at equilibrium all players will either
play 1 or 0. This implies that a player who has beliefs μi,x and who is aware
of the entire process, can follow the same reasoning we do and conclude that
the probability of nature choosing 1 is, in fact, either 0 or 1, rather than
pbi,x . To accommodate these players, define pbi,x as the player’s naive beliefs,
and the beliefs that result from our analysis — as the player’s sophisticated
beliefs. Due to strategic to strategic complementarities, an act that is op-
timal with respect to the naive beliefs will also be optimal with respect to
the sophisticated beliefs.
It is important to note that if naive beliefs were to be ignored, and
players were to have only sophisticated beliefs, then any assignment of 0 or
1 to the games Gx1 , ..., GxJ could be a consistent set of equilibrium beliefs.
However, such a model would not describe the process by which beliefs are
formed. The naive belief formation process is the topic of the next section.3

3 Belief formation process

Our approach to the belief formation question is history- and context-dependent.
Specifically, we assume that games of the type Gx above are being played
over and over again, by different populations [0, 1], and for different values
of x. The history of similar games played in the past, which is assumed to be
common knowledge, determines the beliefs pbx of the individuals in question.
More concretely, we assume that time is discrete and that the game Gx
tie-breaking rule requires some additional assumption about the law of large numbers
applying to a continuum of i.i.d random variables.
3
For a discussion of modeling the formation of rational beliefs, see Gilboa, Postlewaite,
and Schmeidler (2010).
 8

is played in every period by a new generation of players. We further assume

that at the beginning of each period t nature selects a value for xt ∈ X ≡
{x1 , ..., xJ } in an i.i.d. manner, according to a known discrete distribution.
Thus the process is determined by a probability vector (p1 , ..., pJ ).
³ ´
Let Ht = (xτ , ν τ )t−1
τ =1 be the history at the beginning of period t,
where, for τ < t, xτ ∈ X denotes the game played at period τ , and ν τ ∈
{0, 1} denotes its outcome. In each period t, all the players are assumed
to observe the same history Ht . Before playing, they observe the game Gxt
and form an expectation on the probability of a success that is based on
the similarity between the current game and previous games that ended,
respectively, with a success or a failure.

Let there be two matrices of non-negative numbers s+ , s− : X ×X → R+

with the following interpretation. s+ (xτ , xt ) measures the degree of support
that a past game xτ , resulting in ν τ = 1, gives to the outcome 1 at the
new game xt . Similarly, s− (xτ , xt ) measures the degree of support that
a past game xτ , resulting in ν τ = 0, gives to the outcome 0 at the new
game xt . These degrees of support generate naive beliefs as follows. Denote
St = {τ < t | ν τ = 1} and Ft = {τ < t | ν τ = 0}, and set
X
s+ (xτ , xt )
τ ∈St
pb (Ht , xt ) = X X (2)
s+ (xτ , xt ) + s− (xτ , xt )
τ ∈St τ ∈Ft

Note that if the functions s+ , s− are identically 1, the expression above is

simply the relative frequency of 1’s in the history Ht . The formula (2) allows
different past games to have different weight in the evaluation of probabilities
at the current period. Thus, it can be viewed as a generalization of empirical
frequencies to weighted empirical frequencies.4
4
The idea of generating beliefs in a game based on past empirical frequencies is at the
heart of "fictitious play", dating back to Robinson (1951). Extending empirical frequen-
cies to similarity-weighted empirical frequencies was suggested and axiomatized in Billot,
Gilboa, Samet, and Schmeidler (2005), and Gilboa, Lieberman, and Schmeidler (2006).
 9

Specifically, we assume that games with a lower index x are commonly

perceived as a-priori more likely to result in 1 than are games with a higher
index x0 . (This is in line with the assumption that the difference in the
expected payoff of strategies 1 and 0 is strictly decreasing in x.) Thus, a
result of 1 in Gx is less surprising than the same result in a game Gx0 . Hence,
a result of 1 in Gx lends weaker support to the same result in the current
game than would the result 1 in a game Gx0 . Formally, assume that s+ (x, y)
is strictly increasing in its first argument and strictly decreasing in its second
argument. Similarly, we also assume that s− (x, y) is strictly decreasing
in its first argument and strictly increasing in its second argument. An
implication of these assumptions is that pb (Ht , xt ) is strictly decreasing in
its second argument.
The formula (2) is not well-defined for the first period, t = 1. Also,
it allows pb(Ht , x) to be 0 or 1, if history contains only 0-outcomes or one
1-outcomes, respectively. We find such extreme beliefs unwarranted. Hence
we use Equation (2) only when history contains both -outcomes or one 1-
outcomes. Formally, we assume that t ≥ 3, and that history contains at
least one 0-outcome and at least one 1-outcome, so that pb(Ht , x) ∈ (0, 1).

4 The Dynamic Process

We now wish to study the dynamic process in which at every stage t ≥ 1, xt
is drawn from X = {x1 , ..., xJ } according to probabilities (p1 , ..., pJ ), beliefs
are formed in accordance with equation (2), and the players’ behavior in
Gxt is chosen by the beliefs pbt (·).
A state of the process is fully summarized by a matrix of relative fre-
quencies

Here we extend the notion of similarity-weighted empirical frequencies to incorporate di-

rectional thinking.
 10

x = x1 x = x2 ... x = xJ
Rt = 1 rt,11 rt,12 ... rt,1J
0 rt,01 rt,02 ... rt,0J

where rt,ij is the relative frequency, up to time t, of periods in which the

game was Gxj and the outcome was i.
Consider the following matrices:

x = x1 x = x2 ... x = xJ
R0 = 1 p1 0 ... 0
0 0 p2 ... pJ
x = x1 x = x2 ... x = xJ
R1 = 1 p1 p2 ... 0
0 0 0 ... pJ

We can finally present our main result.

Theorem 1 For given p̄ and s+ , s− , there exist distributions (p1 , ..., pJ )

such that there is a positive probability that Rt converges to R0 and a positive
probability that it converges to R1 .

It will be obvious from the proof of the theorem that there is nothing ex-
ceptional about the distributions (p1 , ..., pJ ) that allow convergence to either
of the extreme outcomes. The main condition will be that the probabilities
of the extreme games, p1 , pJ be strictly positive but small, relative to the
other probabilities (and given the values of xk , the similarity functions, and
the function ϕ(·)). In particular, the set of distributions (p1 , ..., pJ ) contains
open sets.

5 Conclusion
Ever since the early days of game theory, there has been a quest for a solution
concept that would satisfy existence and uniqueness, with robustness and
dynamic stability as additional desiderata. The attempt to narrow down
 11

the class of potential predictions was motivated by the desire to make game
theory more meaningful and powerful, whether interpreted descriptively or
normatively. Clearly, even if uniqueness of equilibria cannot be obtained,
tighter theoretical predictions would make the theory more useful, and will
thereby reduce the need to resort to extra-theoretical reasoning in order to
select an equilibrium as a likely or a recommended outcome.
The literature on refinements of Nash equilibrium (see van Damme, 1983)
is generally perceived as falling short of pinpointing unique equilibria in
games. However, the more recent literature, viewing a game in the context
of similar and related games, have resulted in several results that changed
the way we think about equilibrium selection (Carlsson and van Damme,
1994, Burdzy, Frankel, and Pauzner, 2001). These results may suggest that,
in a sufficiently detailed model, a unique equilibrium prediction would exist.
The present paper is offered as an example, showing that incorporation
of additional details into the model may leave the game theoretic prediction
ambiguous. We believe that game theoretic analysis is extremely useful, but
that, in general, it cannot subsume the need in historical and institutional
knowledge. Rather, the formal, mathematical analysis needs to be combined
with such knowledge to generate trustworthy predictions.

6 Appendix: Proof of theorem 1.

First, observe that the relative frequencies of the columns of Rt are governed
only by the selection of x, and are independent of the players’ behavior.
Under our assumptions, for every history Ht we can predict the outcome
of the game played at time t by considering the difference pb(Ht , xt ) − p̄xt
(decreasing in xt ). If this difference is strictly positive, all players’ expecta-
tion pbt (·) will be above the critical belief p̄xt . They will therefore all play 1,
and Nature will select ν t = 1 with probability 1. Otherwise, all players will
play 0 and Nature will select ν t = 1 with probability 1.
 12

Recall that
X ¡ ¢
s+ xτ , xk
³ ´
τ ∈St
pb Ht , xk = X X .
s+ (xτ , xk ) + s− (xτ , xk )
τ ∈St τ ∈Ft

The assumption that t ≥ 3, and that history contains at least one 0-

outcome and one 1-outcome implies that pb(Ht , xt ) ∈ (0, 1). This in turn
implies that the difference pb(Ht , xt ) − p̄xt is strictly positive at xt = x1 and
strictly negative at xt = xJ .
We simplify notation by defining
X ³ ´
Akt = s+ xτ , xk (3)
τ ∈St
X ³ ´
Bkt = s− xτ , xk (4)
τ ∈Ft
³ ´ Akt
pb Ht , xk = (5)
Akt + Bkt
zkt = (1 − p̄xk ) Akt − p̄xk Bkt (6)

so that
pb(Ht , xk ) − p̄xk > 0 ⇔ zkt > 0.

Given that Akt is strictly decreasing in xk , Bkt is strictly increasing in xk ,

and p̄x is strictly increasing in xk , it follows that zkt is strictly decreasing
in xk , hence for history Ht there exists a unique yt ∈ {x1 , ...xJ−1 } for which
zyt t > 0 ≥ zy0 t for any y 0 in X such that y 0 > yt .
At time t, the expected change in zkt is given by:
£ ¤
E zk(t+1) − zkt |yt = y
= E [(1 − p̄xk ) Ak(t+1) − p̄xk Bk(t+1) |yt = y ] − [(1 − p̄xk ) Akt − p̄xk Bkt ]
= (1 − p̄xk ) E [Ak(t+1) − Akt |yt = y ] − p̄xk E[Bk(t+1) − Bkt |yt = y ]
X ³ ´ X ³ ´
= (1 − p̄xk ) pj s+ xj , xk − p̄xk pj s− xj , xk .
xj ≤y xj >y
 13

This is increasing in y, as s+ (·) and s− (·) are nonnegative. Also, it is

decreasing in xk because p̄x and s− (·) are increasing in xk and s+ (·) is
decreasing in xk .
Consider histories Ht that contain at least one 1-outcome and a suffi-
ciently long list of 0-outcomes such that z2t ≤ 0. Such histories have positive
probability as long as p1 , pJ > 0. Since zkt is strictly decreasing in xk , z2t ≤ 0
implies zkt ≤ 0 for k ∈ {2, ..., J}.
The expected change in z2t is given by:
£ ¤ ¡ ¢ X ¡ ¢
E z2(t+1) − z2t |z2t ≤ 0 = (1 − p̄x2 ) p1 s+ x1 , x2 − p̄x2 pj s− xj , x2 .
j≥2
£ ¤
Let (p1 , ..., pJ ) be such that E z2(t+1) − z2t |z2t ≤ 0 < 0. (That is, assume
that p1 > 0 is small enough relative to the other pk ’s.)
£ ¤ £ ¤
Since E zk(t+1) − zkt |yt = y is decreasing in xt , E z2(t+1) − z2t |z2t ≤ 0 <
£ ¤
0 implies E zk(t+1) − z2t |zkt ≤ 0 < 0 for k ∈ {2, ..., J}. We argue that,
given that z2t ≤ 0, there is a positive probability that z2τ ≤ 0 for all
τ > t. To see this, observe that, as long as z2τ ≤ 0 for τ > t, z2τ fol-
lows a Markov process. The distribution of z2τ conditional on z2τ > 0 is
not guaranteed to be stationary. However, if we replace it by any station-
ary distribution, we obtain a new process {ẑ2τ }τ >t that is Markovian, with
£ ¤
E ẑk(t+1) − ẑ2t |ẑ2t ≤ 0 < 0 and that is identical to {z2τ }τ >t as long as the
latter is non-positive. Since {ẑ2τ }τ >t has a positive probability of never be-
coming positive, so does {z2τ }τ >t . This completes the proof that our process
has a positive probability of converging to R0 .
A symmetric argument shows that there are probabilities (p1 , ..., pJ ) for
which the process has a positive probability of converging to R0 . Moreover,
following the arguments above it is clear that one can find such probabilities
for which both events occur with positive probability: basically, one has to
guarantee only that p1 , pJ > 0 are small enough relative to the other pk ’s.
¤
 14

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