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\begin{center}
{\large {\bf Comment on Tullock, Hechter, and Wildavsky }\\[0pt]
}

\bigskip  Eric Rasmusen \\[0pt]

Published: {\it Rationality and Society} (January 1992) 4: 83-94.\\[0pt]

{\it Abstract}\\[0pt]
\end{center}

Game theory has been criticized as neglecting key aspects of individual
behavior and as relying too heavily on special assumptions. It can, in fact,
handle individual heterogeneity if the modeler is willing to carefully
specify how people are different, but to the extent that such things as
heterogeneity and culture are important, the desire for a single unified
model is impossible to satisfy. At the same time, game theory's approach is
very useful for building specialized models.

\noindent  Draft: 2.1 (Draft 1.1, June 1990) \newline

{\small \ \noindent \hspace*{20pt} 2000: Eric Rasmusen, Professor of
Business Economics and Public Policy and Sanjay Subhedar Faculty Fellow,
Indiana University, Kelley School of Business, BU 456,  1309 E 10th Street, 
Bloomington, Indiana, 47405-1701.  Office: (812) 855-9219. Fax:
812-855-3354. Erasmuse@indiana.edu. Php.indiana.edu/$\sim$erasmuse. }

%---------------------------------------------------------------  
\newpage

A theme common to the three criticisms of game theory in this issue of {\it %
Rationality and Society} is that it is a  constricted theory, relying too
heavily on narrow assumptions to be useful in explaining how the world
works. Small changes in assumptions produce big changes in conclusions, and,
in particular, game theory has trouble with the heterogeneity of human
beings---their differences in tastes, information, and culture.

The game theorist's reply must be, I think, that game theory's building
blocks do apply to a wide variety of situations, and are, in fact,
particularly well-suited to heterogeneity, but that the situations being
modelled do not lend themselves to general theories. To the extent that
heterogeneity is important, models ought to be narrow, for a general model
is effectively saying that situations are homogeneous. If situations
apparently similar are actually different, then tailoring the game to the
facts is better than relying on a single game for all facts. In a good
theory, at least some seemingly minor details make a difference, for how
else are we to discover that anything but the obvious is important?

To make this concrete, let us follow Gordon Tullock's good example and
analyze a particular game. Tullock uses the following matrix:

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In its stark form, this matrix conjures up players in a laboratory, whose
behavior, while analyzable, has no intrinsic interest for us. But to call
the properties of a  bare mathematical matrix unrealistic is not a
well-posed statement; to build a model, something must exist to be modelled,
something whose essence is to be captured. Starting with just the matrix, we
are in the position of the RAND game theorists in the early 1950's who were
perplexed by  a certain two-by-two matrix that  generated perverse results.
Albert Tucker, on being asked to give a talk on game theory to the Stanford
psychology department, decided to attach a story to the numbers. The result
was the Prisoner's Dilemma, and a deeper understanding than the mathematics
alone could give (Straffin, 1980).

The Tullock matrix is not a prisoner's dilemma, but a story can nonetheless
be attached to it. Let us try discussing it as a model of the conflict
between offense and defense in a football game.\footnote{%
If your response is:``He's using a game to model a game! '', mentally
substitute a military conflict between offense and defense where the attack
can come on the right or left flank.}  The offensive team is trying to
decide between passing the ball and running with it, and the opposing team
must decide whether to set up a defense against passing or running. The
offensive team would like to do the unexpected, since that is the way to
advance the ball towards the goal. But more is at stake with passing than
with running---5 yards instead of 1, if the numbers above are retained.%
\footnote{%
These numbers are not quite realistic, though their ordinal rank fits
football.  Partly this is a matter of the units of measurement. If the
numbers in the matrix are doubled and one yard is added to each entry, they
become more realistic. Such a change will not affect the optimal strategies
in the slightest--- it is just a change of measurement units.} Let us call
this ``The Football Game.'' Its Nash equilibrium is in mixed (random)
strategies. The Offense passes with probability 1/6, and the Defense uses a
running defense with probability 1/2.\footnote{%
To calculate these, use the fact that in a mixed strategy equilibrium a
player is indifferent between his pure strategies. The expected payoffs for
the Offense from his two pure strategies are $\pi_o(pass) = 5 \gamma -
5(1-\gamma)$ and $\pi_o(run) = -1 \gamma +(1)(1-\gamma)$. Equating these
yields $\gamma = .5$. Similarly, equating $\pi_d(running)=-5 \theta + (1)
(1-\theta)$ and $\pi_d(passing)=5 \theta + (-1) (1-\theta)$ yields $\theta =
1/6$.}

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After discussing the characteristics of the original matrix, Tullock objects
that the players should be, like most people, risk-averse. Gaining five
yards may not please as much as losing five yards displeases--- at least if
one's team is currently ahead in the game. This means that the numbers in
Fiure 2 are no longer valid, for they no longer represent the
payoffs---utility--- but an instrumental means to those payoffs---yards. 
The obvious solution is to change the numbers to fit risk-averse payoffs.
But how can this be done? It requires knowledge of how much each player
values winning the overall game, and how much the gain and losses of yardage
affect winning.

The difficulty of measuring possible payoffs is a common criticism of game
theory, but  empirical work in any subject runs into measurement problems,
and game theory presents no special difficulties---only the standard, hard,
ones. The same problem faces the economist who is asked how many more
oranges John will buy if the price of apples rises. About both fruit and
football, he can make an informed guess using general knowledge and ten
minute's thought, or he can use a government grant and three years' work to
come up with a somewhat better answer. In the absence of a grant (and three
years), let us make a guess, based on the assumption that the Offensive
player is already ahead on points, and replace our earlier matrix with
Figure 3 (I have also added variable $c$, which will be explained shortly):

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\end{center}

The game is no longer zero-sum, but the Nash equilibrium can still be
calculated. It will depend on the term $c$, which is added to answer another
of Tullock's questions: what if the row player does not like playing his
upper action? For the Football Game, the question can be made more concrete:
what if the Offensive team captain becomes jealous when his pass receivers
get attention from the crowd, and suffers disutility pangs of $c$ when he
uses the passing strategy? Jealousy is easy to incorporate; that is what $c$
depicts. The mixed-strategy equilibrium is for the Offence to pass with
probability $\theta=4/15$ and the Defence to use a running defence with
probability $\gamma=c/15 + 3/5$.\footnote{%
To calculate this, use the fact that in a mixed strategy equilibrium a
player is indifferent between his pure strategies. The expected payoffs for
the Offensive player from his two pure strategies are $\pi_o(pass) = (3-c)
\gamma + (-8-c)(1-\gamma)$ and $\pi_o(run) = -3 \gamma +(1)(1-\gamma)$.
Equating these yields $\gamma = c/15 + 3/5$. Similarly, equating $%
\pi_d(running\; defence )=-3 \theta + (3) (1-\theta)$ and $\pi_d(passing\;
defence)=8 \theta + (-1) (1-\theta)$ yields $\theta = 4/15$.}  This yields
the interesting prediction that if the Offensive captain comes to hate his
receivers more ($c$ increases), he will not pass any the less, but the
Defence will use the passing defence less often, knowing that the Offense is
reluctant to take advantage of the opportunity. Thus, not only can risk
aversion and preferences for certain actions be incorporated into the game,
but the model can make predictions about how behavior varies depending on
those parameters.

I have already mentioned that measurement is difficult for the modeller, but
it is is also difficult for the players, a distinctly different objection.
So far the Football Game has assumed that they know each others' utility
functions. Is this justified? On a general level, it seems that  individuals
do act as if they know something about each others' utility functions. When
a businessman opens up a shoe factory, it is under the belief that he knows
pretty well how much other people value shoes--- enough so that in tight
competition, he can still make a normal profit. He may be mistaken, but he
is willing to bet on his knowledge, and since the information is very
important to him, he has invested some effort in acquiring information (even
more, perhaps, than a scholar writing an article on the subject). So the
question is really what happens if the players know each others' payoff
functions, but imperfectly.

A complete model requires specification of what it is that the players know
and do not know.  In the Football Game, let us see what happens if the
Offensive captain knows how jealous he is, but the Defensive captain does
not. Assume that (a) the Defensive captain does not know the exact value of $%
c$, and attaches equal probabilities to $c=0$ and $c=1$; (b) the Offensive
captain knows that the Defensive captain has those beliefs; and (c) in
actual fact, $c=1$.\footnote{%
If the Offensive captain is uncertain about the Defensive captain's beliefs,
that too can be incorporated into the model, at the cost of extra
complexity. If he is certain but wrong, it is even simpler to incorporate.}
In equilibrium, the Offensive player will pass with probability $%
\theta_{c=1}=0$ and the Defensive player uses a running defence with
probability $\gamma=3/5.$ The equilibrium also specifies the behavior that
the Offensive player would adopt if $c=0$; it is to pass with probability $%
\theta_{c=0}=8/15$.\footnote{%
The equilibrium is calculated as follows. Whatever value of $\gamma$ is
picked, it cannot be the case that the Offensive player would mix for both $%
c=0$ and $c=1$; in one case or the other he will use a pure strategy. In
computing the equilibrium with known $c$, it was established that if the
value of $\theta$ was greater than 4/15, then $\gamma=1$. Hence, it cannot
be that either $\theta_{c=0}=1$ or $\theta_{c=1}=1$, or the average would
exceed 1/2 and we would have $\gamma=1$. Since it is the player with $c=1$
who is more reluctant to pass, it must be that $\theta_{c=1}=0 $, and $%
\theta_{c=1} \in (0,1)$. If $\gamma$ is to be between zero and one, it must
be that the average value of $\theta$ equals 4/15, so $4/15 =
.5(\theta_{c=1}) + .5(0)$, which yields $\theta_{c=1}=8/15.$ Since $\gamma$
must be chosen to make the player with $c=0$ willing to mix, it is chosen
using the formula $\gamma=c/15 + 3/5$ found in the earlier equilibrium,
which since $c=0$ gives $\gamma=3/5.$} From the point of view of the
Defensive player, who does not know the value of $c$, the probability of the
Offensive player passing is $\overline{\theta} = .5(0) + .5 (8/15) = 4/15$.
The change in assumptions has changed the Offensive player's behavior, from $%
\theta=4/15$ to $\theta_{c=1}=0$. But this what we should expect: the
Offensive player should pass less often when facing an opponent who
overestimates his incentive to pass. Thus, game theory easily adapts to
differences in tastes and knowledge, but by the very fact that it does so,
predicting different outcomes under different circumstances, it tells us
that a perfectly general theory is impossible.

Even if one accepts that game theory can accommodate different sorts of
utility functions and beliefs, however, one might still quarrel with the
very idea of the mixed-strategy equilibrium--- as, indeed, Tullock does. Not
only do mixed strategies involve ``carefully random'' behavior, but the
equilibrium is weak in the sense that each player is indifferent between at
least two of his pure strategies or he would not be willing to mix between
them. Yet if the equilibrium is to exist, it seems this indifferent player
must carefully pick just the right probabilities for each action.

A mixed-strategy equilibrium does not, however, actually require any
randomization. What it requires is that the mixing player's actions {\it seem%
} random to the other players, whether this results from literal
randomization or not. A football captain might not throw dice in the huddle
, but he surely wishes to take actions unpredictable to the other team. If
the captain chooses his plays based on a deterministic device such as
whether the time remaining in the game is an odd or even number, the game
theorist is justified in modelling the choice as random if it seems random
to the Defense. The captain may even decide in advance to pass on the first
play of the game with probability one, but if the other team believes that
in general only proportion $\theta$ of captains pass on the first play, the
result is the mixed-strategy equilibrium.

The Harsanyi (1973) explanation for mixed strategies cited by Tullock is
similar in flavor, but based on player heterogeneity. Harsanyi suggests that
there are always small features of the situation which would push the
deciding player to one or the other pure strategy, but which cannot be
observed by the other players.  The Football Game with unknown $c$ hints at
this, because what to the Defensive player seems randomization with a 4/15
probability is actually probability 8/15 for one type of Offensive player ($%
c=0$) and probability 1 for the other type ($c=1$). If there were a
continuum of types from $c=0$ to $c=2$, practically all types would be using
pure strategies; but to the Defensive player, who cannot observe $c$, it
would seem that the Offensive player was randomizing. The differences in
types need not even be large; there could be a continuum of types from $c=.98
$ to $c=1.02$, and while they would all be choosing one pure strategy or the
other,they would appear to the Defensive player (and the modeller) be
randomizing.

It should be kept in mind, too, that a mixed-strategy equilibrium is still a
Nash equilibrium: no player can profitably deviate from his assigned
strategy, even though he may be indifferent about it.  This is illustrated
by Tullock's other example, the Hillman-Samet rent-seeking game. In that
game, two players simultaneously offer bribes to an official who will grant
100 dollars to the highest bidder, but keep both bribes.  In the
mixed-strategy equilibrium, the two players randomize their choices of
bribes between 0 and 100. Tullock asks what happens if the game is repeated,
and a third player enters and bids 90 each time. This new strategy will do
very badly. If the original two players fail to react, they too will do
badly, but that does not make the ``Bid 90'' strategy rational unless the
entrant is malicious. If the original players do react, they can achieve
higher payoffs than the entrant by mixing between bids of 0 and bids on the
interval between 90 and 100.\footnote{%
Let $F(x)$ be the cumulative probability that a player bids up to $x$. In
equilibrium, the payoffs from the pure strategies are equal, and the pure
strategies are postulated to be 0 and the $[90,100]$ interval.. The payoff
from bidding 0 is 0, and the probability of winning with a bid of $x >90$ is
the probability $F(x)$ that the other rational player has bid less than $x$,
so $\pi(0) = 0 =\pi(x) = -x+ 100F(x)$, and $F(x) = x/100$. This implies that
a player bids 0 with probability .9, and spreads the rest of his probability
over the interval $[90,100]$. The ``Bid 90'' player will win with
probability $.9^2$, so his expected payoff is $.81(100)-90 = -8.9$.} They
will not be driven from the game; often each will bid 0, sometimes one will
bid high and win, and sometimes both will bid high and one will lose his bid
without reward. The example only shows that rational players should adapt
their behavior to the behavior of irrational ones, not that irrational ones
do better.\footnote{%
In other contexts, however, such as bargaining, irrationality can be a
positive advantage. See Rasmusen, 1989, Chapter 10.)}

I have spent so much time on the Football Game because it illustrates the
sort of sensitivity analysis that is useful for honing intuition. Where
changing the assumptions makes a difference, it should make a difference,
and shows that apparently minor assumptions are not so minor. This approach
places the burden on the modeller to describe the game carefully, but that
burden is inescapable under any method of analyis; a car that moves along
fine on a solid highway will spin its wheels uselessly when you try to drive
it on sand.

Let me turn now to the other two papers. Wildavsky says that game theory
ignores culture, and that culture is important, because ``without a
supportive cultural context, no strategy makes sense.'' The rational man of
game theory is not the reasonable man of law; he is a sociopath, without
preexisting values or relations to defend. In particular, the prisoner's
dilemma is not nearly as useful as has been claimed, since most people are
actually not sociopaths. People in different cultures play the prisoner's
dilemma differently, and a given situation will be a prisoner's dilemma in
one culture but not in another.

The issue comes down to what is similar about humans and what is different.
The position of game theory is that everyone,  whatever their culture, is
best analyzed as a rational maximizer, but what is rational depends on the
particular preferences and constraints available to the individual. To quote
O'Rourke (1988, p. 4), ``A Japanese raised in Riyadh would be an Arab. A
Zulu raised in New Rochelle would be a dentist.'' Japanese, Arabs, Zulus,
and dentists are all rational actors, even if we who are outside their
particular cultures do not share their tastes and beliefs.

As Wildavsky says, one physical situation might call for different models
for different cultures, or even within one culture, if preferences differ.
The Football Game's variants--- the basic game and the variants with risk
aversion, jealousy, and asymmetric information--- all were based on the same
physical situation. Game theory does treat people as if they had preexisting
values; the payoffs are literally values, and the model takes them as
preexisting. Wildavsky hits the point precisely when he says the modeler's
position should be ``Tell me how individuals understand their situation in
terms of their preferences, and I will then model the strategic aspects of
this situation...'' Once provided with the rules of the game---players,
payoffs, actions, and initial beliefs--- game theory predicts the outcome.
If the rules of the game are those of the prisoner's dilemma, the outcome is
that both players will confess, whatever their culture may be. But if people
in the culture enjoy prison, or prefer confession to lying, then the payoffs
are different and the game is no longer a prisoner's dilemma. This is not a
matter of how the game is analyzed, but of how its rules are specified.

Culture theorists may object to the entire idea that people respond to
incentives, believing instead that their responses are preconditioned. This
certainly makes for generality, since it leads to predictions independent of
the parameters of the particular situation. But it seems much like an
extreme form of the Tullock objection that a player may not enjoy playing
one of his actions, which, as seen above, is easily handled by game theory.
The objection must then become that the theory is unimportant, because the
answer is determined by the empirical measurement of the disutility of the
action. That is unobjectionable; theory and empirical investigation each
have their place.

In some contexts, however, one must wonder whether ``culture'' really just
refers to situations with different payoff matrices.  Wildavsky's cultural
groups---hierarchists who trust authorities, individualists who trust market
exchange, egalitarians who trust voluntary groups, and fatalists who trust
no one--- might be identical human beings facing different incentives.
Consider, for example, a game in which Smith must decide whether to buy a
computer from Jones, and Jones must decide whether to sell him a working
computer or a broken one. Smith will be a fatalist and not buy the computer
if there is no enforcement mechanism penalizing Jones for selling a broken
one. Smith will be a hierarchist, and buy, if Jones has a boss who will
punish him for selling broken equipment. Smith will be an individualist, and
buy, if warranty law would force Jones to replace a defective computer.
Smith will be an egalitarian, and buy, if he and Jones are members of the
same university department and Jones's reputation will be ruined if he
defrauds Smith. Although these are not all prisoner's dilemmas, and each
story changes the game, the prisoner's dilemma is the basic building block
for them all. One must only remember that a house requires more than just
the basic concrete blocks.

\bigskip

Finally, I turn to Michael Hechter's discussion of the repeated prisoner's
dilemma. Essentially, he has two objections: that the game has multiple
equilibria and that cooperation is difficult to achieve under incomplete
information. The first objection is quite valid, and often forgotten. The
Folk Theorem tells us that a huge variety of outcomes can occur in
equilibria of the infinitely repeated prisoner's dilemma, including $%
Cooperate$ each period and $Fink$ each period. Arguments have been put
forward for why cooperation might be more likely, but the debate is still
very much alive.\footnote{%
Technical notes, however: (1) The strategy of tit-for-tat is {\it not} a
credible (perfect) equilibrium strategy, because there is insufficient
incentive to punish a $confess$ deviation with $Confess$ the next period.
See Kalai, Samet \& Stanford (1988). (2) Taylor (1990) is cited as claiming
that for large but not too large discount rate there is a unique efficient
equilibrium. In fact, (Always $Confess$) remains an equilibrium outcome for
any discount rate for which (Always $Cooperate$) is an equilibrium outcome.
(3) The repeated prisoner's dilemma is not a game of perfect information. It
includes simultaneous moves, so information sets are not singletons. See
Rasmusen (1989), Chapter 2. (4) Discounting hurts for cooperation, rather
than enhancing it.}

That cooperation is more difficult when information is incomplete, however,
is dubious. Incomplete information, in fact, is the most widely accepted
explanation for why cooperation might ensue. Kreps, Milgrom, Roberts, \&
Wilson (1982) show that even if the prisoner's dilemma is only finitely
repeated (and {\it a fortiori} if it is infinitely repeated), the unique
equilibrium outcome can be cooperation until close to the end of the game.
The extra assumption they add is a small probability that one player (let us
call him Smith) is not rational--- Smith plays the tit-for-tat strategy
whether it helps him or not. The rational player knows of this possibility,
and so he cooperates till near the end of the game. If Smith is indeed
irrational, the other player need not fear an unprovoked confession; but
even if Smith is rational, it is to his advantage to pretend to be
irrational till near the end of the game. The argument is subtle, but the
conclusion is simple: uncertainty over the player's payoffs can make
cooperation more likely, not less.\footnote{%
For elaboration, see Chapter 5 of Rasmusen (1989) .} Under complete
information, cooperation is difficult, so muddying the waters can hardly
hurt.

\bigskip

Hechter and Wildavsky would probably agree with Tullock when he says ``My
objection to game theory is as a formal body of mathematics allegedly
applying to human action, not as a heuristic which makes it easier to think
about certain problems.'' Game theory should indeed be a branch of
storytelling, not of mathematics--- storytelling with the $i$'s dotted and
the $t$'s crossed. One can tell stories that are consistent, and stories
that are not; and formalism helps spot the inconsistencies by forcing the
storyteller to tell one story at a time. When Tullock says that  ``The
problem is that parties with no knowledge of formal game theory are likely
to go through the same process as the trained game theorist,'' he pays a
complement to formal game theory, for economists have learned only slowly
that participants in the marketplace are often wiser than scholars. Long
experience, inherited tradition, or the careful deliberation that
self-interest motivates often lead to behavior whose usefulness is not
apparent to the outsider. But theory has an advantage similar to that of the
factory worker over the craftsman: the theorist can make do without
experience, tradition, and self-interest, and use superior capital to
produce a product that is cheaper and more uniform, if perhaps not so
reliable. Let us not be Luddites; without theoretical tools, only the born
craftsman is able to produce decent scholarship.

A criticism common to all three critics is that game theory relies too
heavily on situation-specific assumptions. We all would prefer generalizing
theory to exemplifying theory, a model of what {\it must} happen instead of
what {\it can} happen, as Franklin Fisher (1989) puts it in his own critique
of game theory. The danger in this is that the modeler may try to force-fit
a model to situations for which it is unsuited, as Wildavsky says is done
with the prisoner's dilemma.  But sociology, political science, and
anthropology are in a wonderful position to escape this danger. These
disciplines have been heavily data-driven, with many descriptive studies of
particular situations, which is the empirical analog of the exemplifying
style of game theory. Game theory can explain not only what happened in a
case study, but what might have happened had conditions been different and
what will happen if the parameters change. If game theory is used in this
way, I think that the critics will be happier.

\pagebreak

\begin{center}
{\bf REFERENCES}
\end{center}

Fisher, Franklin. 1989. Games economists play: a noncooperative view. {\it %
Rand Journal of Economics} 20: 113-124 .

Harsanyi, John. 1973. Games with randomly disturbed payoffs: a new rationale
for mixed strategy equilibrium points. {\it International Journal of Game
Theory.} 2: 1-23.

Hillman, A. and D. Samet. 1987. Dissipation of contestable rents by small
numbers of contenders. {\it Public Choice} 54: 63-82.

Kalai, Ehud, Dov Samet and William Stanford. 1988. Note on reactive
equilibria in the discounted prisoner's dilemma and associated games. {\it %
International Journal of Game Theory} 17: 177-186.

Kreps, David, Paul Milgrom, John Roberts, \& Robert Wilson. 1982. Rational
cooperation in the finitely repeated prisoners' dilemma. {\it Journal of
Economic Theory} 27: 245-52.

O'Rourke, P.J. 1988. {\it Holidays in hell}. New York: Atlantic Monthly
Press.

Rasmusen, Eric. 1989. {\it Games and information}. Oxford: Basil Blackwell.

Rasmusen, Eric. Forthcoming. Folk theorems for the observable implications
of repeated games. {\it Theory and Decision}.

Straffin, Philip. 1980. The prisoner's dilemma. {\it UMAP Journal} 1: 101-3.

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