GAMES
AND
ECONOMIC
BEHAVIOR
4, 218-231 (1992)
Equilibrium Refinement in Psychological Games*
VAN KOLPIN
Department
of Economics,
University of Oregon, Eugene, Oregon 97403-1285
Received December 4, 1989
Psychological games are structures in which a player’s “belief-dependent emotions” may affect realized utility. We show that these mechanisms can be modeled as conventional games, thus traditional game-theoretic intuition is applicable.
Equilibrium refinements are investigated. When formulated so that beliefs reflect
sensitivity to the “trembles” of competitors, perfect and even proper equilibria of
psychological games are shown to exist. Journal ofEconomic Literature Classification Number: 026. o 192 Academic press, ~nc.
1.
INTRODUCTION
Psychological games are structures in which realized utility depends on
emotional reactions to actual outcomes. These emotions may in turn be
influenced by a priori expectations or beliefs; consequently, no single
utility function characterizes a player’s preferences over the physical
outcome set. Indeed, precisely the same choices could be made in two
scenarios, yet if the parties involved harbored distinct expectations, the
utility experienced could be radically different. Games such as these were
recently introduced in an important contribution by Geanakoplos et al.
(1989), henceforth referred to as GPS. Their model and its associated
equilibrium concept compose a new and effective framework in which to
analyze such phenomena. However, this framework does not determine a
game in the usual sense, leading the authors to conclude that “the traditional theory of games is not well suited to the analysis of such beliefdependent psychological considerations. . , .” We suggest an alternative
perspective in the formulation of psychological games and the refinement
* The author is grateful for the helpful suggestions of a referee and an associate editor.
218
0899-8256192 $3 .OO
Copyright 6 1992 by Academic Press, Inc.
All rights of reproduction in any form reserved.
PSYCHOLOGICAL
GAMES
219
of their equilibria. This new approach reveals how conventional gametheoretic insight can be used to attain a better understanding of the intricacies involved in psychological games. Furthermore,
the equilibrium
refinements we propose are in harmony with either modeling approach in
addition to realizing several intuitively pleasing properties which are absent from the original GPS formulations.
In the original GPS model, a player’s beliefs are an artifact of the
model’s equilibrium concept and are beyond self-control. The resulting
structure is gamelike, yet it is clearly not a game in the conventional
sense. We pursue a somewhat different approach by modeling beliefs as if
they were a component of individual choice and extending preferences to
encompass beliefs as well as physical outcomes. One’s initial reaction
may be to object to the dubious assumption that players choose their own
beliefs. However, it should be emphasized that the approach we take does
not require players to literally choose their beliefs, only that they behave
as if they do so. The result is a conventional game which we show to be
behaviorally equivalent to the GPS model.
As psychological games bear a distinct resemblance to finite games, it is
natural that their analysis is enhanced by equilibrium refinements similar
to those suggested by Selten (1975), Myerson (1978), and Kreps and
Wilson (1982). GPS constructs a perfect equilibrium concept for psychological games. Adotping Selten’s (1975) analogy of the “trembling hand,”
a GPS perfect equilibrium is robust to trembles in physical strategies,
while beliefs are fixed and perfectly insulated from such trembles. Unfortunately, this concept fails the test of existence, forcing the authors to
consider weaker equilibrium refinements. We suggest an alternative perspective by defining perfect equilibria in such a way that beliefs, as well as
strategies, reflect sensitivity to the trembles of competitors. We extend
proper equilibria (Myerson, 1978) to psychological games in a similar
fashion and establish general existence results for both of these refinements.
Substantial conceptual differences between the GPS perfect equilibrium set and perfect equilibria as defined here would seem to prohibit
persistence of any generic similarities. This conjecture appears to be further supported by examples revealing that neither set need contain the
other. It is interesting to note that despite these facts, the equilibrium
concept we propose is generically a refinement of the GPS version. This
conclusion is realized with the aid of well-known generic results found in
Kreps and Wilson (1982).
For convenience, the formal results to follow are posed in terms of
strategic form games. The implications of our analysis on extensive form
games are briefly outlined in Section 3.
220
VAN
2.
KOLPIN
THE MODEL AND RESULTS
Let us begin the formal section of this paper by presenting the essential
components of psychological games as defined in GPS. The set of players
is represented by N = (1, . . . , n}. Each player’s set of pure strategies
(actions) is denoted by the finite set Ai and the set of mixed strategies
defined on Ai is Si. Thus if si E Si then si: Ai + [0, l] and Xsi(ai) = 1,
where the sum is over all ai E Ai. We denote the spaces of collective pure
and mixed strategies respectively by A = XiE~Ai and S = XiE~Si. A
player’s beliefs are also an essential factor in psychological games. GPS
devotes considerable attention to formally developing each player’s space
of beliefs. The precise structure of these spaces may be critical when
focusing on particular applications; however, such detail is irrelevant to
the issues we raise. It is only important that each individual’s belief space
be characterizable by a Hausdorff space; that is Vi E N, Bi is a topological
space such that V distinct bi, qi E Bi 3 disjoint neighborhoods of bi and qi,
respectively. Each player’s utility is assumed to be a continuous function
of the strategies pursued by all players and the beliefs held by the individual at hand, Ui: Bi x A + R. It is straightforward to extend these functions
tO Ui: Bi X S +
R by Ui(bi, S) = 2 uEAPs(abi(bi,
a),
where
p,(a)
=
IIiENSi(ai). While only one’s own beliefs directly affect utility, for notational purposes it is convenient to trivially extend Ui to B x S Vi E N by
letting Ui(b, S) E Ui(bi, S) V (6, S) E B X S.
The final component required to complete the GPS formulation of a
psychological game is the “coherent response” mapping for player beliefs. As with the space Bi, the precise form of this mapping is inessential
to our analysis. We simply assume that Vi E N the coherent response map
is a continuous function pi: B-i X S-i + Bi, where B-i = XjEN\{i}Bj and
S..j We bow to notational convenience once again and exS-i
=
xjCN\{i}
tend pi to B X S Vi E N by letting pi(6, S) = pi(b-i, S-i) V (b, S) E B X S,
that is pi is independent of player i’s beliefs and strategies. The intended
interpretation is that fii(b, s) represents the beliefs player i would hold if
she knew b-i were the beliefs held by others and that s-i were the strategies they would be pursuing. We let p = (Pi)ieN.
As an aside, those familiar with the GPS paper will note that p as
defined above differs from the formulation found in GPS. The GPS version is derived from the fixed point of the map p(*, s): B + B, which exists
Vs E S by assumption. In other words, they instead consider the map
0: S + B, where Vs E S, p(s) = p@(s), s). Both versions prove useful in
our analysis.
A strategic form psychological game l? is defined by (ui, pi, Si)iEN with
components satisfying the properties previously described. A psychological equilibrium is a vector of strategies s* E S and beliefs b* E B such that
PSYCHOLOGICAL
GAMES
221
6” = P(b*, s”) and ui(b*, s*) 2 u;(b*, S” \Si) VS~E Si 3 where s* \si = (s*i 3 si)
E S. For a fixed collection of beliefs b E B, u;(b, .): S + R Vi E N,
thereby effectively transforming F into a conventional strategic form
game which we refer to as T(b). Adopting this notation, a psychological
equilibrium is a pair (b*, s*) E B x S for which b* = P(b*, s*) and s* is a
Nash equilibrium of T(b*).
It would seem the endogenous specification of coherent beliefs via j3
distinguishes psychological games as quite different animals from games
in the usual sense; as a consequence, traditional theory and technique
may be of little use in this setting. The pivotal role played by /3 in the
equilibrium concept appears to add even more credence to this claim.
However, an alternative perspective reveals that such mechanisms can
effectively be modeled as more traditional games.
Let us assume, for the moment, that beliefs are chosen, not endogenously imposed. Each i E N has “belief” preferences over this choice
which we assume can be characterized by a continuous function Vi: B x
S + [w with the following properties.
(1) Ui(b\P;(by s), S) > Ui(b\qi, S) V (b, S) E B X S and Vqi E Bi such
that qi f Pi(by s).
(2) ui(b, S) = Ui(b’, s’) V (b, s), (b’, s’) E B X S such that bi = bi and
Pi(b, $1 = pi(b’, ~‘1.
The first condition simply states that coherent beliefs are strictly preferred to noncoherent beliefs, thus pi is the “best response belief map.”
The second condition ensures that “utility”
depends only on the belief
player i chooses and which belief is coherent. (Recall that pi is independent of bi and Si.) For instance, if (Bi, di) is a metric space then one such
function could be Ui(b, S) = -di(bi, Pi(b, s)).
Given that our intent is to design a strategic structure functionally
equivalent to psychological games as defined in GPS, we assume that
preference for coherent beliefs is of the highest priority. That is each
player’s overall preferences are lexicographical in ui and ui. Formally,
given i E N and (b, s), (b’, s’) E B x S, player i prefers (6, s) to (b’, s’) if
and only if ui(b, S) > ui(b’, s’) or ui(b, S) = ui(b’y s’) and ui(b, S) 2 Ui(b’, s’).
The specification of a traditional game is now complete. For each i E N,
the player’s strategy space is Bi x Si and preferences over B X S are
determined by the lexicographical ordering based on (Vi, ui). We refer to
the game we have constructed above as a traditional form of the original
GPS psychological game.
Some readers may find the assumption that beliefs are chosen as far
removed from reality. While this may be true in many circumstances,
useful insight can still be gained from its adoption. Consider the following
222
VAN KOLPIN
game introduced by GPS. Player 1 is to give player 2 one of two gifts,
flowers or chocolates, i.e., Al = {f, c}. Player 2 has no active role, A2 =
{0}, but has beliefs about the probability that the gift will be flowers; B2 =
[0, 11 and &(6, s) = s,(f). Player 1 in turn has beliefs about player 2’s
beliefs; B1 = [O, 11 and Pr(b, s) = bZ. Finally, player 2 is equally happy
with either gift regardless of beliefs and player 1 experiences joy if player
1 believes the gift was a surprise; u2(b, a) = 1, u,(b, a) = 1 - b, if al =f,
and u,(b, a) = b, if aI = c. It is easy to see there is no psychological
equilibrium in pure strategies despite the fact that only one player has an
“active” role, a seemingly paradoxical result. However, player 2’s beliefs
implicitly affect player l’s utility via the model’s equilibrium. Modeling
the structure as a true game explicitly recognizes this factor, making this
formerly counterintuitive
result seem perfectly natural. GPS offers several other examples yielding “peculiar” equilibrium sets. This peculiarity
dissipates when one adopts the perspective that beliefs are “chosen.” For
the purpose of gaining insight into the nature of equilibria, it is not so
important whether or not players literally choose beliefs, but rather that
equilibrium behavior is as if they do so. We formalize the fact that these
traditional forms personify the “behavioral
spirit” of psychological
games with the following proposition.
PROPOSITION
1. Let r, be a psychological game and I7 a traditional
form of r, . Then (b*, s*) is a Nash equilibrium of r if and only if (b*, s*) is
u psychological equilibrium of r, .
Proof.
Suppose (b*, s*) is a Nash equilibrium of I, then Vi E N, (b?,
ST) must be the belief strategy combination which maximizes preferences
given (b?i, s?;). Thus bT must maximize ui and subsequently SF maximizes u;. Formally, vi(b*, s*) 2 ui(b*\bi, s*) Vbi E Bi and ui(b*, s*) 2
ui(b*, S* \si) VSi E Si. But the first condition is equivalent to saying pi(b*,
s*) = b:, thus the set of Nash equilibria of I is identical to the set of
Q.E.D.
psychological equilibria of I,.
In light of the previous result, we henceforth refer to both the GPS
formulation and its traditional form as psychological games. One may
either assume that pi, or equivalently Ui, are given as initial data Vi E N.
(If the functions ui are given, pi may be interpreted as the best response
belief map. So that the parallels between the GPS formulation and our
own persist, we continue to assume pi is a single valued continuous function Vi E N.) Given a psychological game, we shall also use the terms
psychological equilibrium and Nash equilibrium synonymously.
The issue of equilibrium existence in a psychological game I may be
resolved in several ways. One method is to assume fi is continuous and
essentially duplicate the existence proof found in GPS. A slightly more
straightforward
method suggests itself when Bi is compact and convex
PSYCHOLOGICAL
GAMES
223
Vi E N. One may construct a new game, Iz, with 2n players. Players i =
1P--.3 IZ have strategy sets Si and utility functions Ui while Vi = 1, . . . ,
n, player rz + i has Bi as a strategy set and Ui as a utility function. It is
trivial to show that if (b, s) is a Nash equilibrium for I2 it also specifies an
equilibrium in the original psychological game I. To show that I2 has an
equilibrium, one need only construct the best reply correspondences and
appeal to Kakutani’s Fixed Point Theorem in the usual fashion. These
arguments support the following result.
THEOREM
continuous,
1. Zf the belief spaces are compact and convex or if0
then the psychological game r has a Nash equilibrium.
is
It is a well-known fact that Nash equilibria are frequently supported by
what may be referred to as “incredible threats.” The desire to eliminate
such “implausible”
equilibria has generated a wealth of literature on
equilibrium refinements, e.g., Selten (1975), Myerson (1978), Kreps and
Wilson (1982), and van Damme (1983). Equilibrium
refinements are
equally relevant to the study of psychological games. As a consequence,
our analysis now turns to the formulation of perfect and proper equilibria
for psychological games as well as proof of their existence.
Let us first consider the manner in which these equilibrium refinements
were designed in standard strategic form games. The general idea of perfect equilibria is that when choosing a best response to other players’
strategy choices one will not completely ignore strategies these players
propose to choose with zero probability. Intuitively, one instead assigns a
“positive but infinitesimal”
probability to such strategies when determining a best response. As an example, consider the game in Fig. 1 and
denote the row and column choices by {rr , r2} and {c, , Q}, respectively.
The strategy combination (rz, c,) is a Nash equilibrium but it is not perfect. While r2 is a best response to cl , if the row player feels there is even
an infinitesimal probability of c2 being played, the only best response
would be r-1. The equilibrium (t-1, c,) does not suffer this drawback and is
O,l
zo
a0
a0
FIGURE 1
224
VAN KOLPIN
thus a perfect equilibrium. We attempt to preserve the spirit of this definition when defining perfect equilibria for psychological games. As we also
study proper equilibria it is convenient to follow Myerson (1978) in defining perfect equilibria as the limit of e-perfect equilibria.
DEFINITION 1. Given a psychological game I and E > 0, an e-perfect
equilibrium is a pair (6, s) E B x S such that s is totally mixed and the
following properties are satisfied Vi E N.
(1)
(2)
bi = /+(b, S) or, equivalently,
ui(b, s) 2 ui(b\qi, s) Vqi E Bi.
If ui(b, s\ai) < ui(b, .~\a!) then si(ai) 5 E.
A perfect equilibrium is a pair (6, s) E B X S such that Vk = 1, . . . , CC
3ek > 0 and an &perfect equilibrium (bk, sk) such that lim &k = 0 and lim
(b“, sk) = (6, s).
Every E-perfect equilibrium specifies totally mixed physical strategies
so that no feasible choices may be completely ignored. Nonoptimal
choices are, however, assigned small levels of probability.
Beliefs, or
belief strategies depending on the perspective, are in turn coherent with
these perturbations or trembles. As perfect equilibria are limited points of
&-perfect equilibria, they capture the notion that an infinitesimal probability is attached to a competitor’s nonoptimal choice.
It is well known that perfect equilibria of finite strategic games are also
Nash equilibria. The obvious question arises as to whether or not this
result holds true for psychological games as well. We find the answer to
be affirmative via the following theorem.
THEOREM 2. Every perfect
also a Nash equilibrium.
equilibrium
of a psychological
game is
Proof.
Suppose (b, s) is a perfect equilibrium of a psychological game
l?.Let{$>OIk=
1,. . . , x} be a monotone decreasing sequence such
that lim ck = 0 and suppose lim (b”, sk) = (6, s), where (bk, sk) is an ckperfect equilibrium Vk = 1, . . . , ~0. We need to show that p(b, s) = b and
that given any i E N, if si(ai) > 0 for some ai E Ai then ui(b, s\ai) 2 ui(b,
S\Ui)
VU; E Ai.
It is assumed that /3 is continuous 3 lim /3(bk, sk) = /3(b, s). But Vk, (bk,
sk) is an Ek-perfect equilibrium implying p(bk, sk) = bk. Thus /3(b, s) = lim
p(bk, sk) = lim bk = b.
Let i E N be given and suppose si(aJ > 0, for some ai E Ai. Let 6 =
si(aJ/2 and pick K such that Vk 2 K, ck < 6 and I$(ai) - Si(ai)l< 6. It
follows that sf(aJ > ck Vk 1 K. Recalling that (bk, sk) is an Ek-perfect
equilibrium, this implies ui(bk, sk\ai) 2 ui(bk, sk\al) VU: E Ai. By the
continuity of U; we conclude ui(b, s\ai) 2 ui(b, s\al) Val E Ai. Q.E.D.
PSYCHOLOGICAL
GAMES
225
This result verifies that the set of perfect equilibria is included in the
Nash equilibrium set of any given psychological game. Of course this
inclusion may also be strict. This fact is well known for finite strategic
form games, e.g., the example in Fig. 1. As such games are special cases
of psychological games, where each ui is independent of B, it clearly
follows that the inclusion may also be strict for psychological games.
Thus the perfect equilibrium concept does indeed constitute a nontrivial
refinement of the Nash equilibrium concept for psychological games. We
temporarily postpone the question of existence to formalize the notion of
proper equilibria in this context.
Myerson (1978) introduced a further refinement of Nash equilibria
known as proper equilibria. Heuristically, as is the case of perfect equilibria, agents do not completely ignore any of their competitor’s feasible
actions. Moreover, progressively less “weight” is associated with competitor’s nonoptimal actions as they become progressively less optimal.
In the context of psychological games this can be formalized as follows.
2. Given a psychological game I and E > 0, and ~-proper
is a pair (b, s) E B x S such that s is totally mixed and the
properties are satisfied Vi E N.
DEFINITION
equilibrium
following
(1)
(2)
bi = pi(b, S) or, equivalently, ui(b, s) 2 Ui(b\qi, s) Vqi E B;.
If u;(b, s\ai) < ui(b, ,~\a;) then si(ai) I &si(ai).
A proper equilibrium is a pair (b, s) E B x S such that Vk = 1, . . . , x
3~~ > 0 and an Ek-proper equilibrium (bk, sk) such that lim ek = 0 and lim
(bk, sk) = (6, s).
It is obvious from the definitions involved that every proper equilibrium
is also perfect. It is well known that this inclusion may be strict for
standard strategic games, implying the same may be true for psychological games. To ensure that we are not proposing refinement to the empty
set, an existence theorem is in order. We modify Myerson’s proper equilibrium existence proof for ordinary finite strategic form games to yield a
proper equilibrium existence theorem for psychological games. As every
proper equilibrium is also perfect, this verifies existence of perfect equilibria as well.
THEOREM 3. Zf JTis a psychological
game and belief spaces are compact and convex (or if fi is continuous) then there exists at least one
proper equilibrium.
Proof.
We first prove the theorem under the assumption that Bi is
compact and convex Vi E N. Suppose V’E E (0, 1) 3 an E-proper equilibrium.Let{Ek>O1k=l,...,
a}, lim sk = 0, and let (bk, sk) be an .@proper equilibrium Vk. B x S is compact so there is no loss in generality in
226
VAN KOLPIN
assuming lirrr (bk, sk) = (6, s) for some (6, s) E B x S, thereby verifying
proper equilibrium existence. Consequently, our proof is complete if we
verify existence of e-proper equilibria Ve E (0, 1).
Let m = max{lAil Ii E N}, where IAil represents the cardinality of the
set Ai. Given E E (0, 1) let 6 = Em/m. V i E N define S7 = {Si E Si 1si(ai) 2 6
V ai E Ai} and let S” = X IeNS:. We construct an “a-proper reply”
correspondence [F;: B x S” + Bi X Sf as follows.
Fi(b, S) = {(qi, tJ E Bi X SF 14; = p;(b, S) and Ui(b\qi, ~\a;) < Ui(b\qi, ,~\a;)
3
ti(Ui)
S: Eti(Ul)
V Uiy
Ul' E
A;}.
Given any (b, s) E Bi X SF, [Fi(b, s) contains ordered pairs with (1)
identical belief coordinates and (2) physical strategy coordinates satisfying a finite collection of linear inequalities, implying IFi(b, s) is closed and
convex. Let qi = /3(b, S) and VU; E Ai let p(ai) = [{u; E A;1 ui(b\q;, s\ai) <
ui(b\qi
, s\u~)}[.
By letting ti(ai) = Ep(aJl(~e~(a’:)),where the sum is over all
uf E A;, it follows that (qi, t;) E ff;(b, s) and thus IFiis not empty-valued.
Both Ui and pi are assumed continuous so that 1F;is an upper hemicontinuous correspondence.
Let iF = X iev[Fi; thus, IF: B X Se * B X S” and
satisfies all conditions of Kakutani’s Fixed Point Theorem which implies
3 (b”, sE)E B x S” such that (6”, s&) E lF(b”, s&). In other words (b”, 9) is an
E-proper equilibrium of the psychological game I.
Essentially the same proof may be used when fi is assumed continuous.
The only change is in the definition of Fi. Given E E (0, 1) one defines Fi:
S: + Si” by
=
{t; E
Sf
1 u;@;(S),
s\ai)
<
U;@;(S),
S\Ul)
3
ti(UJ
5
Eti(ui)
v Ui,
uf
E Ai}.
Q.E.D
While proper equilibria are not discussed in GPS, perfectness is examined. They offer a formulation quite different from our own. (We use the
term GPS perfect equilibrium to distinguish the concept from that presented in Definition 1.)
DEFINITION
3.
Given a psychological game I, (6, s) E B x S is a GPS
if (b, s) is a Nash equilibrium of I and u is a perfect
of the conventional game T(b).
perfect equilibrium
equilibrium
Definitions
1 and 3 characterize “hyper-rationality”
from different
viewpoints. Each may seem reasonable. depending on one’s perspective.
For several reasons, however, we tend to favor the version of perfectness
introduced in Definition 1. We elaborate as follows.
One drawback to the GPS perfect equilibrium concept is that existence
PSYCHOLOGICAL
GAMES
zo
-b,, 1
FIGURE
227
2
cannot be guaranteed. Consider the game depicted in Fig. 2, the strategic
form of a game found in GPS. The utilities listed are those corresponding
to the functions Ui: B x A -+ [w, i = 1, 2, where 1 is the row player and 2
the column player. Thus, for instance, ui(b, (Y,, c,)) = -b, and ~~(6,
(r,, ci)) = 1 (player 2’s utilities are not belief dependent). To complete specification of the psychological game, assume B1 = B2 = [0, 11,
PI@, $1 = b2, and Mb, s) = sdn). F or each k 2 2 let ck = 4/(k - 1) and
define (bk, sk) by bt = 6; = 3/(k - 1) = s!(~i), s&c,) = 1 - l/k. It is easy to
show (bk, sk) constitutes an ek-perfect equilibrium; furthermore, lim (bk,
sk) = (b, s), where 6, = b2 = 0 = s,(YI), SZ(CI) = 1. Thus (6, s) is a perfect
equilibrium for the given psychological game. However, there are no GPS
perfect equilibria in this game as (b, s) as constructed above can be shown
to be the only psychological equilibrium and s is not a perfect equilibrium
in I’(b).
A second reason we favor Definition 1 is somewhat more subjective. If
players attach infinitesimal “trembles”
to opponents’ strategies, it seems
an extension of Selten’s original definition “ought” to require that both
strategies and beliefs reflect some sort of sensitivity to such trembles.
Such “belief sensitivity” is not inherent to GPS perfect equilibria, but is
an integral part of the perfect equilibrium formulation found in Definition 1.
Finally, we point out that GPS perfect equilibria can be plagued by
forms of “irrationality”
which are avoided by perfect equilibria. Consider
the psychological game depicted in Fig. 3, utilities corresponding to ui:
B x A + I!& i = 1, 2. To complete specification of the psychological game,
let BI = {0}, B2 = [0, 11, /3,(b, s) = 0, and /32(b, s) = s,(Y,). If (b, s) satisfies
6, = 0, b2 = 1, S,(Q) = 1, and sz(cl) = 1 then it is easy to show that (b, s) is
a GPS perfect equilibrium. This outcome seems somewhat “unsatisfactory” as its perfectness is entirely based on unshakeable beliefs that
player 1 will choose r, with absolute certainty. However, if trembles are
anticipated, such certainty is unwarranted. Readers may find Fig. 4, the
228
VAN
KOLPIN
1,1
191
o,o
030
0,b2
4,1
Wp
4,1
FIGURE 3
extensive form of Fig. 3, even more compelling. A game of perfect information is depicted in which player 1 moves first and player 2 controls the
two decision nodes which follow. (See van Damme (1983) for a formal
introduction to games in extensive form.) The GPS perfect equilibrium
constructed above is characterized by player 1 moving left and player 2
moving left at both decision nodes, while player 2 believes with absolute
certainty that player 1 will move left, even in the event that player 2’s
second node is reached.
Let us return to the game in Fig. 2 and again let (b, s) represent this
game’s sole perfect equilibrium. The reader may have noted that if utilities are perturbed slightly, (b, s) is also a GPS perfect equilibrium. Indeed,
V[ > 0 if utilities in the cell (ri , ci) change to (-6, - 4, 1) then s becomes a
perfect equilibrium on P(b). This fact is not conincidental to the particular
game we have constructed. Using the “generic” results of Kreps and
Wilson (1982) we are able to show that “generically”
the set of GPS
perfect equilibria contains the set of perfect equilibria. This result, ap-
o,o
0,b2
FIGURE 4
PSYCHOLOGICAL GAMES
229
pearing formally as Theorem 4, has two important implications: (1) it
reveals that generically the perfect equilibrium concept is a refinement of
the GPS perfect equilibrium concept, and (2) it establishes a generic existence result for GPS perfect equlibria.
Given the action space A = X iENAi define W(A) to be the set of all
games which may be generated from A and a utility function profile. A
single utility function defined on A may be identified with an element of
[wIAl; thus, a utility function profile may be identified with RIAl. As each
game in V(A) is identified with a utility function profile, W(A) may itself
be identified with R”lAl. Let p represent Lebesgue measure on this space.
We may now state the following result which is a corollary to formal
results appearing in Kreps and Wilson (1982).
COROLLARY 1 (Kreps and Wilson, 1982). There exists E C W(A) such
that p(E) = 0 and VI/I E q(A)IE, all Nash equilibria of II, are perfect.
We use this result to show that, generically, any perfect equilibrium (b,
s) of a psychological game P is also GPS perfect. Let B = X iE nBi and A =
X iEnAi be given as belief and action spaces. Let T(B X A) represent the
set of all psychological games that can be generated from B x A. Finally,
let ‘P(B x A) be the set of all conventional games resulting from perfect
equilibrium beliefs in some F E r(B x A), i.e., q(B x A) = {I/J 1 3r E
T(B x A) and (b, s) a perfect equilibrium of F for which $ = r(b)}.
Given any b E B, T(b) E W(A) so clearly V(B x A) C q(A). On the
other hand, it is possible to have a psychological game in which utilities
are insensitive to beliefs; thus, q(A) c W(B x A) and it follows that
q(B x A) = q(A).
THEOREM 4. There exists E C q(B x A), p(E) = 0 such that VT E
T(B x A)for which 3a perfect equilibrium (b, s) E B x S satisfying T(b) E
1Ir(B x A)IE, it follows that s is a perfect equilibrium of I(b).
Proof.
By Corollary 1 3E c W(A) such that p(E) = 0 and QI+!JE W(A)/
E, all Nash equilibria of I/Jare perfect. But if l7 E I’(B x A) and (b, s) is a
perfect equilibrium of F. Theorem 2 implies that s is a Nash equilibrium of
T(b). Thus if I’(b) E q(B x A)IE = Y(A)IE, it follows that s is a perfect
equilibrium of T(b).
Q.E.D.
3.
EXTENSIVEPSYCHOLOGICALGAMES
Those familiar with the GPS paper will note that a majority of its formal
analysis is devoted to psychological games in extensive form. We have
chosen to focus attention on the strategic form primarily for two reasons.
First, we wish to show that a fruitful study of psychological equilibrium
230
VAN
KOLPIN
refinements can be made in the context of the strategic form. This is an
important step as nonexistence of the GPS perfect equilibrium concept
suggests that one must rely on weaker refinements such as sequential and
subgame perfect equilibria where dynamic structures are necessary. Second, the strategic form is a perfectly satisfactory medium in which to
study the issues we raise. Formal introduction of the extensive form
would not only unnecessarily lengthen this manuscript, but also require
additional notation, a potential distraction from the topics under investigation. For those readers with specific interests in extensive form psychological games, we conclude this paper by outlining definitions of sequential and subgame perfect equilibria in the same spirit as our formulation of
perfect equilibria.
Sequential equilibria in conventional games were introduced by Kreps
and Wilson (1982). In brief, a sequential equilibrium is an “assessment” p
which is both consistent and sequentially rational. The consistency criterion is a limit condition somewhat reminiscent of the trembling hand
analogy. A GPS sequential equilibrium for the psychological game I is a
pair (b, p) such that given the coherent beliefs b, the assessment p is a
sequential equilibrium of the conventional game T(b). Consequently, the
assessment is consistent given the beliefs, but the beliefs themselves need
not be in harmony with the limit of “perturbations”
defining consistency.
As an alternative to the GPS formulation, one can define a sequential
equilibrium in a psychological game to be a belief, assessment pair (6, p)
which is sequentially rational and which satisfies a limit condition ensuring mutual consistency of assessments and beliefs. As with conventional
extensive games, it can be shown that in the “psychological”
context, all
perfect equilibria correspond to sequential equilibria (the versions we
propose), but not vice versa.
The reader may begin to see a pattern emerging between the equilibrium refinements offered by GPS and those proposed here. The GPS
approach focuses on the psychological equilibria for which given fixed
coherent beliefs, conventional refinement criteria are satisfied on the resulting conventional
game. The approach taken in this paper is to
reformulate the requisite refinement criteria in terms of the entire psychological game, rather than on a “branch” of the game restricted to fixed
beliefs. These two approaches can also be adopted when defining subgame perfect equilibria. The GPS version is, of course, the set of psychological equilibria (6, s) such that s is subgame perfect in the conventional
game T(b). The version proposed here defines a subgame perfect equilibrium to be a psychological equilibrium which is a psychological equilibrium on every psychological subgame. Intuitively a psychological subgame is the restriction of the game to a subtree such that every
information set intersecting the subtree lies entirely within the subtree
PSYCHOLOGICALGAMES
231
and utilities at endpoints of the subtree are independent of beliefs regarding “what goes on” outside of the subtree. It is interesting to note that
unlike perfect and sequential equilibria which generically refine their GPS
counterparts, the GPS subgame perfect equilibrium set can be shown to
be contained in the subgame perfect equilibrium set as defined above.
However, the set of all subgame perfect equilibria in a psychological game
can be constructed through backward induction; much as in conventional
games, a feature is not shared by the GPS subgame perfect equilibrium
set.
REFERENCES
J., PEARCE, D., AND STACCHETTI,
E. (1989). “PsychologicalGames
and
Sequential Rationality,” Games Econ. Behau. 1, 60-79.
KREPS, D., AND WILSON R. (1982). “Sequential Equilibria,”
Econometrica 50, 863-894.
MYERSON,
R. (1978). “Refinements of the Nash Equilibrium Concept,” Znt. J. Game Theory
7, 73-80.
SELTEN, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Points in
Extensive Games,” Znr. J. Game Theory 4, 25-55.
VAN DAMME, E. (1983). Rejinements of the Nash Equilibrium Concept. Berlin: SpringerVerlag.
GEANAKOPLOS,