Many weaknesses of game theory are cured by new models that

embody simple cognitive principles, while maintaining the formalism and

generality that makes game theory useful. Social preference models can

generate team reasoning by combining reciprocation and correlated equilibrium.

Models of limited iterated thinking explain data better than equilibrium

models do; and they self-repair problems of implausibility and

multiplicity of equilibria

Camerer, Colin F.

English

Caltech Authors - Main

Commentary/Colman: Cooperation, psychological game theory, and limitations of rationality in social interaction 
Colman discusses "Stackelberg reasoning" and "team thinking," 
and he mentions (sect. 8.1, para. 3) that the collective preferences 
of team reasoning can be triggered by the acceptance of a group 
identity in certain contexts. But he doesn't explain where these al-
ternative reasoning methods come from, how they survive, or how, 
if cooperation in social dilemmas is sensitive to the cost/benefit 
ratio, we might "trade-off" the different reasoning methods in 
some meta-reasoning process. Hamilton's (1964) "kin-selection," 
Trivers' (1971) "reciprocal altruism," and Alexander's (1987) "in-
direct reciprocity" models might at least offer a way to think about 
answering these questions. 
If we wish to incorporate the social emotions triggered by a 
strategic choice into our models, how might we proceed? Hollis 
and Sugden (1993) explained (p. 28) why our attitudes toward 
consequences cannot be simply "bundled in" with the existing util-
ities of a game. A more plausible path then may be to alter the 
weighting we attach to the consequences, along the lines of the 
"rank dependent" transformation of the cumulative probability 
distribution, which has worked so well among the alternatives to 
expected utility theory (see Starmer 2000). In this way, some plau-
sible improvements to orthodox game theory might be developed, 
as has already happened to expected utility theOlY in choice under 
risk. 
Behavioral game theory: Plausible formal 
models that predict accurately 
Colin F. Camerer 
Division of Social Sciences, California Institute of Technology, Pasadena, CA 
91125. camerer@hss.caltech.edu http://hss.caltech.edu/-camerer 
Abstract: Many weaknesses of game theory are cured by new models that 
embody simple cognitive principles, while maintaining the formalism and 
generality that makes game theory useful. Social preference models can 
generate team reasoning by combining reciprocation and correlated equi-
librium. Models oflimited iterated thinking explain data better than equi-
librium models do; and they self-repair problems of implausibility and 
multiplicity of eqUilibria. 
Andrew Colman's wonderful, timely, and provocative article col-
lects several long-standing complaints about game theory. Part of 
the problem is that game theory has used lots of applied math and 
little empirical observation. Theorists think that deriving per-
fectly precise analytical predictions about what people will do 
(under differing assumptions about rationality) from pure rea-
soning is the greatest challenge. Perhaps it is; but why is this the 
main activity? The important uses of game theory are prescrip-
tive (e.g., giving people good advice) and descriptive (predicting 
what is likely to happen), because good advice (and good design 
of institutions) requires a good model of how people are likely to 
play. It is often said that studying analytical game theory helps a 
player understand what might happen, vaguely, even if it does not 
yield direct advice. This is like saying that studying physics helps 
you win at pool because the balls move according to physical laws. 
A little phYSics probably doesn't hurt, but also helps very little 
compared to watching other pool players, practicing, getting 
coaching, studying what makes other players crumble under 
pressure, and so on. 
While Colman emphasizes the shortcomings of standard theOlY, 
the real challenge is in creating new theory that is psychological 
(his term) or "behavioral" (my earlier term from 1990; they are 
synonymous). Models that are cognitively plaUSible, explain data 
(mostly experimental), and are as general as analytical models, 
have developed very rapidly in just the last few years. Colman 
mentions some. Others are described in my book (Camerer 2003). 
An important step is to remember that games are defined over 
utilities, but in the world (and even the lab) we can usually only 
measure pecuniary payoffs - status, territory, number of offspring, 
money, and so forth. The fact that people cooperate in the pris-
oner's dilemma (PD) is not a refutation of game theory per se; it 
is a refutation of the joint hypothesis of optimization (obeying 
dominance) and the auxiliary hypothesis that they care only about 
the payoffs we observe them to earn (their own money). The self-
interest hypotheSiS is what's at fault. 
Several new approaches to modeling this sort of "social prefer-
ences" improve on similar work by social psycholOgists (men-
tioned in sect. 8.1), because the new models are deSigned to work 
across games and endogenize when players help or hurt others. 
For example, in Rabin's fairness theory, player A treats another 
player's move as giving herself (A) a good or bad payoff, and forms 
a judgment of whether the other player is being nice or mean. 
Players are assumed to reciprocate niceness and also meanness. 
Rabin's model is a way to formalize conditional cooperation -
people cooperate if they expect others to do so. This prOvides a 
way to anchor the idea of "team reasoning" in methodological in-
dividualism. In experiments on group identity and cooperation, a 
treatment (like subjecting subjects to a common fate or dividing 
them into two rooms) or categorization (whether they like cats or 
dogs better) is used to divide subjects into groups. In the Rabin 
approach, PD and public goods games are coordination games in 
which players are trying to coordinate on their level of mutual 
niceness or meanness. 
Experimental identity manipulations can be seen as correlating 
devices that tell subjects which equilibrium will be played, that is, 
whether they can expect cooperation from the other players or not 
(which is self-enforcing if they like to reciprocate). This explana-
tion is not merely relabeling the phenomenon, because it makes a 
sharp prediction: A correlated equilibrium requires a publicly ob-
servable variable that players commonly know. If identity is a cor-
relating device, then when it is not commonly known, cooperation 
will fall apart. For example, suppose members of the A team ("in-
formed A's") are informed that they will play other A's, but the in-
formed As' partners will not know whether they are playing A's or 
B's. Some theories of pure empathy or group identification pre-
dict that who the other players think they are playing won't mat-
ter to the informed A's because they just like to help their team-
mates. The correlated equilibrium interpretation predicts that 
cooperation will shrink if informed A's know that their partners 
don't know who they are playing, because A's only cooperate with 
other A's if they can expect cooperation by their partners. So there 
is not necessarily a conflict between an individualist approach and 
team reasoning: "Teamness" can arise purely through the con-
junction of reciprocal individual preferences and observable cor-
relating variables, which create shared beliefs about what team 
members are likely to do. What those variables are is an interest-
ing empirical matter. 
Another type of model weakens the mutual consistency of play-
ers' choices and beliefs. This might seem like a step backward but 
it is not - in fact, it solves several problems that mutual consis-
tency (equilibrium) creates. In the cognitive hierarchy (CH) 
model of Camerer et al. (2002), a Poisson distribution of discrete 
levels of thinking is derived from a reduced-form constraint on 
working memory. Players who use 0 levels will randomize. Players 
at level K > 0 believe others are using 0 to K to 1 levels. They know 
the normalized distribution of lower-level thinkers, and what 
those others do, and best respond according to their beliefs. The 
model has one parameter, 'T, the average number oflevels of think-
ing (it averages around 1.5 in about a hundred games). In the CH 
model, every strategy is played with positive probability, so there 
are no incredible threats and odd beliefs after surprising moves. 
Once 'T is fixed (say 1.5), the model produces an exact statistical 
distribution of strategy frequencies - so it is rrwre precise in games 
with multiple equilibria, and is generally rrwre empirically accu-
rate than equilibrium models. The model can explain focal points 
in matching games iflevel-O subjects choose what springs up. The 
model also has "economic value": If subjects had used it to fore-
cast what others were likely to do, and best responded to the 
model's advice, they would have earned substantially more (about 
BEHAVIORAL AND BRAIN SCIENCES (2003) 26:2 157 
CommentarylColman: Cooperation, psychological game theory, and limitations of rationality in social interaction 
a third ofthe economic value of perfect advice). Nash equilibrium, 
in contrast, sometimes has negative economic value. 
Beliefs, intentions, and evolution: Old versus 
new psychological game theory 
Jeffrey P. Carpenter and Peter Hans Matthews 
Economics Department, Middlebury College, Middlebury, VT 05753. 
jpc@middlebury,edu peter,h.matthew8@middlebury.edu 
http://community.mlddlebury.edu/-jcarpent/ 
http://community.middlebury.edu/-pmatthewl 
Abstract: We compare Colman's proposed "psychological game theory" 
with the existing literature on psychological games (Geanakoplos et aI. 
1989), in which beliefs and intentions assume a prominent role. We also 
discuss experimental evidence on intentions, with a particular emphasis on 
reciprocal behavior, as well as recent efforts to show that such behavior is 
consistent with social evolution. 
Andrew Colman's target article is a call to build a new, psycholog-
ical, game theory based on "nonstandard assumptions." Our im-
mediate purpose is to remind readers that the earlier work of 
Ceanakoplos et al. (1989), henceforth abbreviated as CPS, which 
the target article cites but does not discuss in detail, established 
the foundations for a theory of "psychologlcal games" that achieves 
at least some of the same ends. Our brief review of CPS and some 
of its descendants - in particular, the work of Rabin (1993) and 
Falk and Fischbacher (2000) - will also allow us to elaborate on 
the connections between psychologlcal games, experimental eco-
nomics, and social evolution. 
The basic premise of CPS is that payoffs are sometimes a func-
tion of both actions and beliefs about these actions, where the lat-
ter assumes the form of a subjective probability measure over the 
product of strategy spaces. If these beliefs are "coherent" - that 
is, the information embodied in second-order beliefs are consis-
tent with the first-order beliefs, and so on - and this coherence is 
common knowledge, then the influence of second (and higher) or-
der beliefs can be reduced to a set of common first-order beliefs. 
That is, in a two-player psychological game, for example, the util-
ities of A and B are functions of the strategies of each and the be-
liefs of each about these strategles. A psycholOgical Nash eqUilib-
rium (PNE) is then a strategy profile in which, given their beliefs, 
neither A nor B would prefer to deviate, and these first-order be-
liefs are correct. If these augmented utilities are continuous, then 
all normal form psycholOgical games must have at least one PNE. 
The introduction of beliefs prOvides a natural framework for 
modeling the role of intentions in strategic contests, and this could 
well prove to be the most important application of CPS. It is ob-
vious that intentions matter to decision-makers - consider the le-
gal difference between manslaughter and murder - and that game 
theorists would do well to heed the advice of Colman and others 
who advocate a more behavioral approach. 
For a time, it was not clear whether or not the CPS framework 
was tractable. Rabin (1993), which Colman cites as an example of 
behavioral, rather than psychological, game theory, was perhaps 
the first to illustrate how a normal form psychologlcal game could 
be derived from a "material game" with the addition of parsimo-
nious "kindness beliefs." In the standard two-person prisoner's 
dilemma (PD), for example, he showed that the "all cooperate" 
and "all defect" outcomes could both be rationalized as PNEs. 
As Rabin (1993) himself notes, this transformation of the PD is 
not equivalent to the substitution of altruistic agents for self-
interested ones: the "all defect" outcome, in which each prisoner 
believes that the other(s) will defect, could not otherwise be an 
equilibrium. This is an important caveat to the recommendation 
that we endow economic actors with "nonstandard reasoning 
processes," and prompts the question: What observed behavior 
will the "new psychological game theory" explain that an old(er) 
158 BEHAVIORAL AND BRAIN SCIENCES (2003) 26:2 
CPS-inspired one cannot? Or, in narrower terms, what are the 
shortcomings of game theoretic models that incorporate the role 
of intentions, and therefore such emotions as surprise or resent-
fulness? 
The answers are not obvious, not least because there are so few 
examples of the transformation of material games into plausible 
psychologlcal ones, and almost all of these share Rabin's (1993) 
emphasis on kindness and reciprocal behavior. It does seem to us, 
however, that to the extent that Colman's "nonstandard reasoning" 
can be formalized in terms of intentions and beliefs, there are 
fewer differences between the old and new psychological game 
theories than at first it seems. 
There is considerable experimental evidence that intentions 
matter. Consider, for example, Falk et al. (2000), in which a first 
mover can either give money to, or take money away from, a sec-
ond mover, and any money glven is tripled before it reaches the 
second mover, who must then decide whether to give money back, 
or take money from, the first mover. Their analysis suggests that 
there is a strong relationship between what the first and second 
movers do: in particular, the more the first mover gives (takes), the 
more the second mover takes (gives) back. 
Falk et al. (2000) find that first mover giving (taking) is inter-
preted as a friendly (unfriendly) act, and that these intentions mat-
ter. \Vithout the influence of beliefs or intentions on utilities, there 
would be a Single Nash eqUilibrium in which the first mover takes 
as much as possible because she "knows" that the second has no 
material incentive to retaliate. Although this behavior can also be 
supported as a PNE, so can that in which the first mover gives and 
expects a return and the second mover understands this intention 
and reciprocates. When the experiment is changed so that the first 
mover's choice is determined randomly, and there are no inten-
tions for the second mover to impute, the correlation between first 
and second mover actions collapses. We see this as evidence that 
beliefs - in particular, intentions - matter, but also that once these 
beliefs have been incOlporated, a modified "rational choice frame-
work" is still useful. 
Building on both CPS and Rabin (1993), DufWenberg and 
Kirchsteiger (199S) and Falk and Fischbacher (2000) derive vari-
ations of Rabin's (1993) "fairness equilibrium" for extensive form 
games, with results that are consistent with experimental evi-
dence. The Simplest of these is the ultimatum game, in which a 
first mover offers some share of a pie to a second mover who must 
then accept or reject the proposal. With kindness functions simi-
lar to Rabin's (1993), Falk and Fischbacher (2000) show that the 
ultimatum game has a unique PNE that varies with the "reci-
procity parameters" of proposer and responder. Furthermore, this 
eqUilibrium is consistent with the observations that the modal of-
fer is half the surplus, that offers near the mode are seldom re-
jected, that there are few of the low offers that are consistent with 
the subgame perfect equilibrium, and that most of these low of-
fers are rejected. 
This result does not tell us, though, whether this outcome is 
consistent with the development of reCiprocal intentions or norms 
over time, or, in other words, whether social evolution favors those 
with "good intentions." To be more concrete, suppose that the 
proposers and responders in the ultimatum game are drawn from 
two distinct populations and matched at random each period, and 
that these populations are heterogeneous with respect to inten-
tion. Could these intentions survive "selection" based on differ-
ences in material outcomes'? Or do these intentions impose sub-
stantial costs on those who have them? 
There are still no definitive answers to these questions, but the 
results in Binmore et aI. (1995), henceforth abbreviated as BCS, 
hint that prosodal intentions will sometimes survive. BCS con-
sider a "miniature ultimatum game" with a limited strategy space 
and show there are two stable equilibria within this framework. 
The first corresponds to the subgame perfect eqUilibrium - pro-
posers are selfish, and responders accept these selfish offers - but 
in the second, proposers are fair and a substantial share of re-
sponders would tum down an unfair offer. Furthermore, these dy-


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