Camerer, Chong - Behavioral Game Theory - Thinking, Learning And Teaching (2001 - 70 Pages).pdf - Teoria gier - narayana

Behavioral Game Theory:

Thinking, Learning, and Teaching

Colin F. Camerer 1

California Institute of Technology

Pasadena, CA 91125

Teck-Hua Ho

Wharton School, University of Pennsylvania

Philadelphia PA 19104

Juin Kuan Chong

National University of Singapore

Kent Ridge Crescent

Singapore 119260

November 14, 2001

1 This research was supported by NSF grants SBR 9730364, SBR 9730187 and SES-0078911. Thanks

to many people for helpful comments on this research, particularly Caltech colleagues (especially Richard

McKelvey, Tom Palfrey, and Charles Plott), Monica Capra, Vince Crawford, John Du®y, Drew Fuden-

berg, John Kagel, members of the MacArthur Preferences Network, our research assistants and collabora-

tors Dan Clendenning, Graham Free, David Hsia, Ming Hsu, Hongjai Rhee, and Xin Wang, and seminar

audience members too numerous to mention. Dan Levin gave the shooting-ahead military example.

Dave Cooper, Ido Erev, and Bill Frechette wrote helpful emails.

1 Introduction

Game theory is a mathematical system for analyzing and predicting how humans behave

in strategic situations. Standard equilibrium analyses assume all players: 1) form beliefs

based on analysis of what others might do (strategic thinking); 2) choose a best response

given those beliefs (optimization); 3) adjust best responses and beliefs until they are

mutually consistent (equilibrium).

It is widely-accepted that not every player behaves rationally in complex situations,

so assumptions (1) and (2) are sometimes violated. For explaining consumer choices

and other decisions, rationality may still be an adequate approximation even if a modest

percentage of players violate the theory. But game theory is di®erent. Players' fates

are intertwined. The presence of players who do not think strategically or optimize can

therefore change what rational players should do. As a result, what a population of

players is likely to do when some are not thinking strategically and optimizing can only

be predicted by an analysis which uses the tools of (1)-(3) but accounts for bounded

rationality as well, preferably in a precise way. 2

It is also unlikely that equilibrium (3) is reached instantaneously in one-shot games.

The idea of instant equilibration is so unnatural that perhaps an equilibrium should not

be thought of as a prediction which is vulnerable to falsi¯cation at all. Instead, it should

be thought of as the limiting outcome of an unspeci¯ed learning or evolutionary process

that unfolds over time. 3 In this view, equilibrium is the end of the story of how strategic

thinking, optimization, and equilibration (or learning) work, not the beginning (one-shot)

or the middle (equilibration).

This paper has three goals. First we develop an index of bounded rationality which

measures players' steps of thinking and uses one parameter to specify how heterogeneous a

population of players is. Coupled with best response, this index makes a unique prediction

of behavior in any one-shot game. Second, we develop a learning algorithm (called

Functional Experience-Weighted Attraction Learning (fEWA)) to compute the path of

2 Our models are related to important concepts like rationalizability, which weakens the mutual con-

sistency requirement, and behavior of ¯nite automata. The di®erence is that we work with simple

parametric forms and concentrate on ¯tting them to data.

3 In his thesis proposing a concept of equilibrium, Nash himself suggested equilibrium might arise

from some \mass action" which adapted over time. Taking up Nash's implicit suggestion, later analyses

¯lled in details of where evolutionary dynamics lead (see Weibull, 1995; Mailath, 1998).

equilibration. The algorithm generalizes both ¯ctitious play and reinforcement models

and has shown greater empirical predictive power than those models in many games

(adjusting for complexity, of course). Consequently, fEWA can serve as an empirical

device for ¯nding the behavioral resting point as a function of the initial conditions.

Third, we show how the index of bounded rationality and the learning algorithm can be

used to understand repeated game behaviors such as reputation building and strategic

teaching.

Our approach is guided by three stylistic principles: Precision; generality; and em-

pirical discipline. The ¯rst two are standard desiderata in game theory; the third is a

cornerstone in experimental economics.

Precision : Because game theory predictions are sharp, it is not hard to spot likely

deviations and counterexamples. Until recently, most of the experimental literature con-

sisted of documenting deviations (or successes) and presenting a simple model, usually

specialized to the game at hand. The hard part is to distill the deviations into an al-

ternative theory that is similarly precise as standard theory and can be widely applied.

We favor speci¯cations that use one or two free parameters to express crucial elements

of behavioral °exibility because people are di®erent. We also prefer to let data, rather

than our intuition, specify parameter values. 4

Generality : Much of the power of equilibrium analyses, and their widespread use,

comes from the fact that the same principles can be applied to many di®erent games,

using the universal language of mathematics. Widespread use of the language creates a

dialogue that sharpens theory and cumulates worldwide knowhow. Behavioral models of

games are also meant to be general, in the sense that the same models can be applied

to many games with minimal customization. The insistence on generality is common in

economics, but is not universal. Many researchers in psychology believe that behavior

is so context-speci¯c that it is impossible to have a common theory that applies to all

contexts. Our view is that we can't know whether general theories fail until they are

broadly applied. Showing that customized models of di®erent games ¯t well does not

mean there isn't a general theory waiting to be discovered that is even better.

4 While great triumphs of economic theory come from parameter-free models (e.g., Nash equilibrium),

relying on a small number of free parameters is more typical in economic modeling. For example, nothing

in the theory of intertemporal choice pins a discount factor ± to a speci¯c value. But if a wide range

of phenomena are consistent with a value like .95, then as economists we are comfortable working with

such a value despite the fact that it does not emerge from axioms or deeper principles.

It is noteworthy that in the search for generality, the models we describe below are

typically ¯t to dozens of di®erent data sets, rather than one or two. The number of

subject-periods used when games are pooled is usually several thousand. This doesn't

mean the results are conclusive or unshakeable. It just illustrates what we mean by a

general model.

Empirical discipline : Our approach is heavily disciplined by data. Because game

theory is about people (and groups of people) thinking about what other people and

groups will do, it is unlikely that pure logic alone will tell us what they will happen. 5

As the physicist Murray Gell-Mann said, `Think how hard physics would be if particles

could think.' It is even harder if we don't watch what `particles' do when interacting.

Our insistence on empirical discipline is shared by others, past and present. Von

Neumann and Morgenstern (1944) thought that

the empirical background of economic science is de¯nitely inadequate...it

would have been absurd in physics to expect Kepler and Newton without

Tycho Brahe,{ and there is no reason to hope for an easier development in

economics

Fifty years later Eric Van Damme (1999) thought the same:

Without having a broad set of facts on which to theorize, there is a certain

danger of spending too much time on models that are mathematically ele-

gant, yet have little connection to actual behavior. At present our empirical

knowledge is inadequate and it is an interesting question why game theorists

have not turned more frequently to psychologists for information about the

learning and information processes used by humans.

The data we use to inform theory are experimental because game-theoretic predictions

are notoriously sensitive to what players know, when they move, and what their payo®s

are. Laboratory environments provide crucial control of all these variables (see Crawford,

1997). As in other lab sciences, the idea is to use lab control to sort out which theories

5 As Thomas Schelling (1960, p. 164) wrote \One cannot, without empirical evidence, deduce what

understandings can be perceived in a nonzero-sum game of maneuver any more than one can prove, by

purely formal deduction, that a particular joke is bound to be funny."

work well and which don't, then later use them to help understand patterns in naturally-

occurring data. In this respect, behavioral game theory resembles data-driven ¯elds

like labor economics or ¯nance more than analytical game theory. The large body of

experimental data accumulated over the last couple of decades (and particularly the last

¯ve years; see Camerer, 2002) is a treasure trove which can be used to sort out which

simple parametric models ¯t well.

While the primary goal of behavioral game theory models is to make accurate pre-

dictions when equilibrium concepts do not, it can also circumvent two central problems

in game theory: Re¯nement and selection. Because we replace the strict best-response

(optimization) assumption with stochastic better-response, all possible paths are part of

a (statistical) equilibrium. As a result, there is no need to apply subgame perfection or

propose belief re¯nements (to update beliefs after zero-probability events where Bayes'

rule is helpless). Furthermore, with plausible parameter values the thinking and learning

models often solve the long-standing problem of selecting one of several Nash equilibria,

in a statistical sense, because the models make a unimodal statistical prediction rather

than predicting multiple modes. Therefore, while the thinking-steps model generalizes

the concept of equilibrium, it can also be more precise (in a statistical sense) when

equilibrium is imprecise (cf. Lucas, 1986). 6

We make three remarks before proceeding. First, while we do believe the thinking,

learning and teaching models in this paper do a good job of explaining some experimental

regularity parsimoniously, lots of other models are being actively explored. 7 The models

in this paper illustrate what most other models also strive to explain, and how they are

6 Lucas (1986) makes a similar point in macroeconomic models. Rational expectations often yields

indeterminacy whereas adaptive expectations pins down a dynamic path. Lucas writes (p. S421): \The

issue involves a question concerning how collections of people behave in a speci¯c situation. Economic

theory does not resolve the question...It is hard to see what can advance the discussion short of assembling

a collection of people, putting them in the situation of interest, and observing what they do."

7 Quantal response equilibrium (QRE), a statistical generalization of Nash, almost always explains the

direction of deviations from Nash and should replace Nash as the static benchmark that other models

are routinely compared to (see Goeree and Holt, in press. Stahl and Wilson (1995), Capra (1999) and

Goeree and Holt (1999b) have models of limited thinking in one-shot games which are similar to ours.

There are many learning models. fEWA generalizes some of them (though reinforcement with payo®

variability adjustment is di®erent; see Erev, Bereby-Meyer, and Roth, 1999). Other approaches include

rule learning (Stahl, 1996, 2000), and earlier AI tools like genetic algorithms or genetic programming to

\breed" rules. Finally, there are no alternative models of strategic teaching that we know of but this is

an important area others should look at.

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