SOLUTION CONCEPT IN THE BASIC STRATEGIC GAME
by
John Bryant*
Rice University
August, 2014
© 2014
This paper first considers the extension of maximization in simple choice to the basic
strategic game. Iterated elimination of strictly set and (standard) vector dominated strategies are
examined and compared in their roles as solution concepts, exploiting the Prisoner’s Dilemma.
Unfortunately, these dominance criteria typically eliminate only a fraction of the strategies.
Hence attention turns to the overwhelming favorite solution concept for strategic games, Nash
equilibrium, which typically eliminates all but a few strategies. Nash equilibrium is not an
extension of maximization like the dominance criteria, and the use of Nash equilibrium presents
three basic problems, whether Nash equilibrium is indeed self-enforcing in the relevant sense,
why players play Nash equilibrium strategies to begin with, and, if there are multiple Nash
equilibria, how players coordinate on one. The role of the self-enforcing property of Nash
equilibrium in enabling, but not generating, a solution is described, exploiting the Prisoner’s
Dilemma and Bryant’s “min rule” game as examples. Motivated by relations between the
dominance criteria and Nash equilibrium, conditions are provided under which the assumptions
of the basic strategic game might be augmented so that pure strategy Nash equilibrium outcomes
are implied. Finally it is suggested how such augmentations might be developed by expanding
upon well known assumptions in the literature, examples being common knowledge, deviation
avoidance, and, in the case of mixed strategy equilibrium, stability. Given the importance of
Nash equilibrium in game theory, and the importance of game theory for strategic analysis, this
may be a fruitful approach.
Keywords: Game Theory, Strategic Games, Solution Concept, Dominance, Nash Equilibrium
JEL codes: C72
*John Bryant, Dept. of Economics-MS 22, Rice University, P. O. Box 1892, Houston, Texas
77251-1892, USA, [email protected], phone 713-348-3341, fax 713-348-5278
(kindly clearly mark “for John Bryant”), total word count - 8217.
I. INTRODUCTION
Probably everyone who studies Game Theory at one point or another wonders “why
‘solution concept,’ what’s wrong with ‘solution?’” This paper revisits this question, and attempts
to clarify, or at least expound on, the matter. In doing so, full use is made of well known game
theory texts, as excellent sources for widely accepted content, and masters of clarity. Even in the
texts, however, ambiguity over solution versus solution concept can creep in, and perhaps
appropriately so. And this is not a matter of mere semantics, but a deep, substantive issue at the
very core of Game Theory.
Game Theory starts with a conundrum. As it models strategic interaction between
rational, maximizing players Game Theory immediately presents a fundamental conceptual
problem, how to extend maximization from simple choice to strategic interaction, as emphasized
by Von Neumann and Morgenstern themselves (c. 1944, Chapter I, 2. -2.2.4).
“Thus each participant [in a “social exchange economy”] attempts to maximize a function … of
which he does not control all the variables. This is certainly no maximum problem, but a
peculiar and disconcerting mixture of several conflicting maximum problems.
… the conceptual differences between the original ([Robinson] Crusoe’s) maximum
problem and the more complex problem [of a “social exchange economy”]. … we face here and
now a really conceptual -- and not merely technical -- difficulty. And it is this problem which the
theory of games of strategy is mainly devised to meet” (Von Neumann and Morgenstern, 1967
ed., pp. 11, 12).
This “conceptual difficulty” which game theory “is mainly devised to meet” is no small
matter. It may be one of the most fundamental aspects of social interaction, and perhaps of
decentralized systems generally. It also bears on the nature of knowledge and uncertainty in
situations of strategic interaction. More narrowly, it explains the use of “solution concepts” in
game theory. If maximization were directly applicable to strategic interaction then the solution
of the game would just be the maximizations for all the players.
The nature of this conceptual difficulty is best studied in a basic strategic game, which
serves to isolate strategic interaction and the particular problems it poses. This strategic game
assumes a finite number of players, that their respective strategy spaces, consisting of a finite
number of strategies, and their respective payoffs to the cross product of all the players
strategies, or outcomes, are known. These are the only payoffs, and, in particular, payoffs do not
depend upon the structure of the game itself, or anything other than the outcomes. These payoffs
can be in ordinal utility or in Von Neumann and Morgenstern cardinal utility. The players are
isolated, and they are inferring what this game means for their behavior, strategy choice, taking
into account that they all are so inferring, and are known to be rational payoff maximizing agents
with full information on the structure of the game. Then “All players choose simultaneously
and independently of each other a strategy from their strategy sets and the game is
over.” (Aliprantis and Chakrabarti, 2011, p. 53) It is a one shot game. To avoid indeterminacies
caused by indifference, that is by equal payoffs, the assumption is added here that every player’s
payoffs imply a strict order over the outcomes. Finally, it is assumed that the specifications of
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these components comprise a complete description of the relevant environment. We are thinking
of people facing an environment, thinking about it as if conceptualizing a basic strategic game,
and playing a strategy.
These players are not facing a maximization problem because the payoffs depend upon
the cross product of the players’ strategies. Payoffs are jointly determined. Hence the call for an
extension of maximization to strategic interaction for these rational maximizing players. First set
and vector dominance, which are explicit extensions of maximization, are treated. Then, with
the limitations of these approaches in mind, the basis for the use of Nash equilibrium, the most
widely used and fundamental “solution concept ” of game theory, is addressed. Nash
equilibrium does not involve an extension of maximization per se, as do the dominance criteria,
but the exploitation of maximizing behavior is critical to it.
II. DOMINANCE SOLUTION CONCEPTS
Set dominance is the least heroic, and least “powerful,” extension of maximization to
strategic interaction. (Bryant, 1987) Indeed, one could argue that it is not an extension at all.
Consider two strategies, 1 and 2, of player A, say. If the minimum payoff of the set of player A’s
payoffs associated with strategy 1 is larger than the maximum payoff of the set of player A’s
payoffs associated with strategy 2, then strategy 1 set dominates strategy 2. No matter what
happens should he play strategy 1, and no matter what happens should he play strategy 2,
strategy 1 yields a higher payoff. Hence player A drops strategy 2 from consideration. Being
rational maximizers the players can infer the dropping of strategies by each other. Moreover
when strategies, and their associated payoffs, are deleted this can cause set dominant relations
among the truncated remaining strategies to emerge. When this rational processing is complete
there remain strategies none of which are set dominated by another, including possibly a single
strategy, for each player. Usually in the context of (standard) vector dominance, this rational
processing is often referred to as the iterated elimination of dominated strategies.
There are two additional aspects of set dominance worth noting here. In the first place,
set dominance exploits only ranking, so it only requires ordinal utility, not cardinal utility,
payoffs. This may be of significance as one may encounter circumstances in which just ranking
outcomes is problem enough. Indeed cardinal utility may in reality be a very strong assumption
for some contexts. Secondly, set dominance does not exploit the independence of the, isolated,
players’ choices of strategy. It would not influence the player’s behavior, as characterized by set
dominance, if he believed the other players could, or possibly could, make their decisions
contingent on his. This possible dependence also may characterize some circumstances one
encounters. This second point on independence is clarified by turning to (standard) vector
dominance, which does exploit independence.
(Standard) vector dominance is a somewhat more “powerful” extension of maximization
to strategic interaction in that it eliminates all of the strategies eliminated by set dominance and,
typically, more. Again consider two strategies, 1 and 2, of player A. If for every cross product of
strategies of the other players the payoff to strategy 1 is higher than the payoff for strategy 2 for
player A then strategy 1 vector dominates strategy 2. As with set dominance the rational
maximizing players then engage in the iterated elimination of dominated strategies. When this
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rational processing is complete there remain strategies none of which are vector dominated by
another, including possibly a single strategy, for each player.
Vector dominance, as well, exploits only ranking of payoffs, so it also only requires
ordinal utility, not cardinal utility, payoffs. However, the ranking is done for each given cross
product of strategies of the other players separately, and thus exploits independence. For
example, the payoff from strategy 1 and one cross product of strategies of the other players is not
compared to the payoff from strategy 2 and a different cross product of strategies of the other
players, as might be called for if the other players could, or possibly could, make their decisions
contingent on player A’s decision. Thus set dominance implies vector dominance, but not the
reverse. The iteratively non-vector dominated strategies are a subset (not necessarily proper) of
the iteratively non-set dominated strategies. For example consider that famous, and insight rich,
game theoretic conundrum the Prisoner’s Dilemma, in standard matrix form (where 4 is the best
outcome).
FIGURE I
Prisoner’s Dilemma
The (2,2) outcome is associated with vector dominant strategy 1, and is a Nash equilibrium as
well, but it is Pareto dominated by the (3,3) outcome, hence the fame of the Prisoner’s Dilemma.
However, strategy 1 is not set dominant as 3 is preferred to 2, and none of the strategies are ruled
out by set dominance. This illustrates that dominance criteria and Nash equilibrium do not bear a
simple relation with welfare, and portrays the gulf between maximization in a simple choice
setting and strategic interaction. It also illustrates that set and vector dominance are not above
challenge as extensions of maximization to strategic interaction, set dominance for admitting
unreasonable behavior, and vector dominance for eliminating reasonable behavior.
There is a further characterization of vector dominance that sheds a familiar light on the
independence condition, and on ordinal and cardinal payoffs as well. Let us first suppose that the
payoffs are ordinal and not cardinal, and that strategy 1 of player A vector dominates strategy 2
of player A, as in the Prisoner’s Dilemma. Now consider instead corresponding strategies 1* and
2* that are any strategies with Von Neumann and Morgenstern cardinal utility payoffs that
separately and together imply the same strict ranking of the outcomes involving strategies 1 and
2 as do strategies 1 and 2 themselves. Then strategy 1* vector dominates strategy 2*, and, for
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any given probability distribution on the strategies of the other players, player A’s expected
utility of strategy 1* is greater than his expected utility of strategy 2*. Vector dominance may
not be beyond challenge, but it is not defenseless either.
III. THE SOLUTION CONCEPT AND NASH EQUILIBRIUM
The example of the Prisoner’s Dilemma not withstanding, vector dominance is a
demanding criterion to meet. There can remain many strategies and outcomes after the iterated
elimination of vector dominated strategies. This suggests that the assumptions of the basic
strategic game, supplemented with an extension of maximization, may be insufficient to
determine behavior, to logically imply an outcome, in many circumstances. Thus this forcefully
brings up the issue of the solution concept of the basic strategic game. Consider a broad
definition of solution concept advocated by Roger Myerson:
“In general, a solution concept is any rule for specifying predictions as to how players might be
expected to behave in any given game.” (Myerson, 1991, p. 107) “The simplest kind of solution
concept is one that specifies the set of strategies that each player might reasonably be expected to
use, without making any attempt to assess the probability of the various strategies .” (Myerson,
1991, p. 88)
The above should not be interpreted to mean that just any rule should do for “solution
concept,” as Myerson himself suggests. Indeed, one view is that what one should mean by
“solution concept,” or perhaps “solution,” is the logical implications for behavior of the
assumptions of the basic strategic game, as supplemented with an extension of maximization
presumably. If there are no further implications than those derived from the iterated elimination
of vector dominated strategies, then that solution concept is that the players do not play the
eliminated strategies and do play one of their remaining strategies. However, this strategy
choice, among the iteratively undominated strategies, presumably is caused by something, and
we have a violation of the complete description assumption. Of course, it could just be that the
basic strategic game does capture all the relevant strategic elements, but there are non-strategic
elements at work, and perhaps these non-strategic elements are hard to specify. Naturally one
would hope instead that somehow the solution is that one particular unique outcome is implied
by a strategic structure, or, barring that, at least fewer outcomes than those of iteratively
undominated strategies, and hopefully, probability distributions over players’ strategies as
Myerson suggests. There needs to be some “mechanism” beyond what we have seen thus far to
produce this result, some stronger solution concept. Ideally this would involve alterations or
supplements to the assumptions of the basic strategic game and the resulting logical implications
for behavior.
The prime candidate in the literature for solution concept for the basic strategic game is
the Nash equilibrium, which does not, of itself, take this ideal form. This mathematical object,
Nash equilibrium, is even more directly tied to maximization than are dominance criteria, and
sheds light on the nature of the concept of solution concept itself. The role of this solution
concept (and it is not a dynamic object, formally anyway) is a major subject of interest indeed in
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game theory, consider: “Nash equilibrium is the solution concept used to solve simultaneousmove games;” (Dixit and Skeath, c. 2004, p. 113) “The Nash equilibrium will be the basic
concept throughout this book. ... Nash equilibrium (or simply an equilibrium or even a
solution) of a strategic game …” (Aliprantis and Chakrabarti, 2011, pp. 48, 54) “Nash’s (1951)
concept of equilibrium is probably the most important solution concept in game
theory.” (Myerson, 1991, p. 105) and “...the general n-person noncooperative game, presented in
finite strategic form. The fundamental solution concept for such games was formulated by John
Nash.” (Shubik, c. 1982, p. 240). The following discussion treats pure strategy Nash
equilibrium, leaving the consideration of mixed strategies to the end.
A pure strategy Nash equilibrium is an outcome such that, for every player, if the other
players play their Nash equilibrium strategies, then the player in question maximizes his payoff
by playing his Nash equilibrium strategy as well: this is often referred to as Nash equilibrium
being self-enforcing. A Nash equilibrium is an individual best reply strategy for all players. The
player contemplates deviating from the Nash equilibrium and observes that that reduces his
payoff. Thus Nash equilibrium itself is not an extension of maximization, but involves a direct
use of maximization by the players. Still it is in the spirit of vector dominance as a vector
dominating strategy has a higher payoff than the dominated strategy for every given play by the
other players, while the player’s Nash equilibrium strategy has the highest payoff for Nash
equilibrium play by the other players. Moreover, notice that Nash equilibrium does just depend
upon the structure of the game as specified by the basic strategic game, which would seem to
support its status as a solution concept for that game. It seems as though there are no alterations
or supplements to the assumptions of the basic strategic game involved. But how can this be
possible in a situation of strategic interaction? Were Von Neumann and Morgenstern wrong
about their conceptual difficulty after all? Or, alternatively, where is the sleight of hand, if you
will? If the players all expect the others to play their respective Nash equilibrium strategy they
will do so themselves, as rational maximizing agents. But how do they as isolated rational
maximizing agents come to that common expectation as THAT itself is not in general implied by
the structure of the game? At least in the first instance players are pursuing good outcomes not
common expectations. And we have already seen in the Prisoner’s Dilemma that common
expectations and Nash equilibrium need not have good welfare properties, should such properties
be an element in a solution concept. (This approach of Nash equilibrium, by the way, is an
example of a tried and true technique used in mathematics, namely if you do not know how to
solve a problem show how it is really a problem you do know how to solve.)
Nash equilibrium as a solution concept poses three basic problems. Is Nash equilibrium
indeed self-enforcing, in the appropriate sense, and what role does this play? Why do players
play a Nash equilibrium to begin with? Finally, if there are multiple Nash equilibria how do
players coordinate on the same one?
The above characterization of Nash Equilibrium as being self-enforcing is widely
observed and warrants scrutiny. It is worth noting starting off that this payoff reducing
“deviating” is not physically dynamic but a purported thought process by players in the strategic
game. Now suppose the players all imagine all imagining playing their strategy of a, and the
same, Nash equilibrium somehow. Next the individual player is imagining taking as given that
the others will be playing their Nash equilibrium strategies, and is thinking about the
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consequences of deviating. But cannot he then imagine the others considering deviating, and
they the same, and then the possibility of a number of them doing so such that they are not
necessarily made worse off by deviating, this being considered? Such contemplation of others’
deviating does not violate independence. Common expectation may, then, be even more of a
sticky wicket, given the basic strategic game’s rational maximizing players inferring what the
game means for their behavior, strategy choice, taking into account that they all are so inferring,
and freely choosing. On the other hand, such implicit “communal” deviation may seem, to the
individual players, to be a bad bet relative to playing the Nash equilibrium strategy. More
basically, though, imagining being “at” a Nash equilibrium and contemplating deviating is not
actually being “at” a Nash equilibrium, which, after all, can only occur when the “players choose
simultaneously and independently of each other a strategy and the game is over.” (Aliprantis and
Chakrabarti, 2011, p. 53) Self- enforcing Nash equilibrium would not resolve the problem of the
source of common expectations, of why players play a Nash equilibrium to begin with, but rather
it enables the conceivable possibility of a resolution.
For a simple and powerful example, consider again the Prisoner’s Dilemma. Both
players deviating generates an outcome Pareto superior to the Nash equilibrium, but if they knew
the other would deviate they would play their Nash equilibrium strategy. But that does not imply
that they know what the other will do. The Prisoner’s Dilemma particularly clearly exhibits
potential benefits to deviation from a Nash equilibrium. While such deviation is ruled out by
vector dominance, it is not by set dominance. Do, then, people tacitly cooperate in this
circumstance? As to what experimental games reveal about behavior in this conundrum, the
experimental results are mixed. “There is typically a mixture of cooperation [both deviating] and
defection [playing the Nash equilibrium strategy], with the mix being somewhat sensitive to
[cardinal] payoffs and procedural factors, with some variation across groups.” (Holt c. 2007, p.
38). The Nash equilibrium in the Prisoner’s Dilemma not being self-enforcing provides one
possible explanation for these results, including there being mixed results.
There is another, or complementary, explanation for a non self-enforcing Nash
equilibrium in the Prisoners Dilemma. Perhaps people are “socialized” into playing the Pareto
dominating outcome in Prisoner’s Dilemma situations. Such “socialization” would seem a good
idea, in the context of the game anyway, if people can be so “socialized.” Perhaps this is
illuminating the distinction between “narrow self interest” and “enlightened self interest!”
Likely such “socialization” violates the assumptions of the basic strategic game that payoffs
determine behavior and that the payoffs themselves are just determined by the outcomes alone,
and are not influenced by the structure of the game. This source of violation of the assumptions
of the basic strategic game, “socialization” in the Prisoner’s Dilemma, may be a deep insight
into the functioning of societies (and religions?) provided by game theory. To explain the
experimental results it would have to be imperfect or non-uniform socialization. In any case, the
purported self-enforcing property of Nash equilibrium is an important matter bearing on the issue
of the Nash equilibrium as solution concept itself. For is not a non self-enforcing Nash
equilibrium not an equilibrium at all?
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IV. FOUR PROPOSITIONS ON NASH EQUILIBRIUM AS SOLUTION CONCEPT
As a start on further considering the Nash equilibrium as solution concept, first its
relationship to set and vector dominance is addressed. We have already seen in the Prisoner’s
Dilemma that the outcome associated with the vector dominant strategies and the Nash
equilibrium outcome are the same outcome, which certainly is suggestive. The first proposition
generalizes this.
Proposition I: If after the iterated elimination of set or of vector dominated strategies only a
single outcome remains it is a pure strategy Nash equilibrium.
Proof: Suppose this unique iteratively undominated outcome is not a pure strategy Nash
equilibrium. Then for at least one player, given that the others play their iteratively undominated
strategy, the player is better off not playing his iteratively undominated strategy, but another
strategy. Consider the strategy of his which would have the highest payoff for him when the
others play their iteratively undominated strategies. Then that strategy is not dominated and
cannot have been eliminated. (See also Aliprantis and Chakrabarti, 2011, p. 64, and Dixit and
Skeath, c. 2004, p. 96)
This certainly seems supportive of the idea of Nash equilibrium as a solution concept,
taking set and vector dominance as bona fide extensions of maximization to strategic interaction.
But here iterated elimination of dominated strategies yields a common expectation. The isolated
rational maximizing agents do come to that common Nash equilibrium expectation, as here it is
implied by the structure of the game, which is known to them.
Proposition II: Every Nash equilibrium outcome of the original game remains as a Nash
equilibrium outcome after the iterated elimination of set or vector dominated strategies.
Proof: A Nash equilibrium has the property that for every player if the other players play
their Nash equilibrium strategies the player in question gets his highest payoff possible, given
their play, by playing his Nash equilibrium strategy. Then those Nash equilibrium strategies
cannot be dominated and cannot be eliminated. (Aliprantis and Chakrabarti (2011, p. 62) in
“Theorem 2.13 (The Iterated Elimination of Strictly Dominated Strategies)” provide a broader
result for vector dominance).
Notice that Proposition II implies that the Proposition I outcome is a unique Nash
equilibrium outcome in the original game. However, this is a necessary, but not sufficient,
condition for iterated dominance yielding a single outcome.
In terms of supporting Nash equilibrium as a solution concept Proposition II is a mixed
bag. On the one hand, by Proposition II the Nash equilibria are not eliminated by the dominance
criteria, which non-elimination should be true of a solution, taking set and vector dominance as
bona fide extensions of maximization to strategic interaction, once again, anyway. On the other
hand, if there are multiple Nash equilibria, and there well can be, then the dominance criteria do
not reduce this set. This adds a wrinkle to the basic issue of why the players would play a Nash
equilibrium to begin with. With multiple Nash equilibria how can isolated players coordinate on
one of them, and which one? To take a canonical case:
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“To focus the analysis consider the following tacit coordination game, which is a strategic form
representation of John Bryant’s (1983) Keynesian coordination game.” (Van Huyck, Battalio and
Beil, 1990, p. 235) For further discussion see Cooper (1999) and Holt (c. 2007, Chapter 26
“Coordination Games,” pp. 325-337).
U(i) = A { minimum [e(1), ...,e(N)] } - B {e(i)},
i = 1, ...,N; A > B> 0; 0 ≤ e(i) ≤ E.
[Symmetric, Summary Statistic]
(Vincent Crawford first saw parallels between this game and J. J. Rousseau’s Stag Hunt parable,
but the original motivation was the work of J. M. Keynes.)
This specification is of a multiple player multiple strategy team game. Numerous
individuals’ efforts are complements. There are N>1 individuals, labeled as individual 1 through
N, and individual i is one particular one of those N individuals. Each individual i chooses an
“effort level” e(i) that is between 0 and E, E>0. The payoff to individual i is called U(i) and is
increasing in the minimum effort level chosen in the whole group of N individuals, and
decreasing, but less strongly, in the individual’s own effort. This strategic game has a whole
continuum of Pareto ranked strict Nash equilibria, where all individuals pick the same effort
level, at any level between 0 and E. Like the Prisoner’s Dilemma, this graphically illustrates that
common expectations and Nash equilibrium need not have good welfare properties. All but the
highest effort level, Pareto optimal, equilibrium are Pareto dominated by other equilibria. But
here every possible strategy choice of an individual is the strategy choice associated with one of
the Nash equilibria, the dominance criteria eliminate no strategies, and the Nash equilibria are of
linear measure zero in the cross product of strategy sets. In short it is an extreme coordination
problem. Indeed, in Van Huyck, Battalio and Beil’s (1990), replicated and robust, experiments,
with a finite strategy version of this game, a Nash equilibrium is never observed.
While yielding an extreme coordination problem, this case may not be special. To model
team production, where “moving together” beats “going it alone,” with constant returns to scale
and continuity in “effort,” then, by Euler’s theorem, the payoff function must be nondifferentiable, as with the “min rule.” This suggests that such coordination problems, while
extreme, might be prevalent with isolated players, particularly as specialization, and
interdependence generally, grows. For example, interpret the e(i) as component parts production.
Thus the no surplus condition (a condition of Walrasian equilibrium not met in team production)
is not a mere technicality. Indeed such coordination problems may play an important role in
macroeconomic and financial instability as well as in industrial organization, see Russell
Cooper’s (1999) seminal book Coordination Games: Complementarities and Macroeconomics.
At the same time, there is a refinement of the Nash equilibrium solution concept that is
applicable here even though these equilibria are strict. In particular, imperfect information,
global games, versions of this model provide interesting insights into the possible elimination of
the continuum of equilibria and the coordination problem. The non-differentiability of the “min
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rule” is critical to the continuum of equilibria in this game, as it means that team production
obtains even for infinitesimal incremental increases in effort. In the very interesting global
games “min rule” paper of Carlsson and Ganslandt (1998) the realized effort levels are noisy,
which “smooths out” the kink so that differentiability obtains, and the continuum of equilibria is
eliminated. On the other hand, in the also very interesting global games “min rule” paper of
Asheim and Yoo (2008), the uncertainty is in individual specific payoff parameters, analogous to
A above, which maintains the “min rule” kink, and the continuum of equilibria and coordination
problem remains. Thus the choice of how to introduce uncertainty is not innocuous, and the
environment being modeled may have implications for it. Further, this refinement does not
address the basic issue of why players play a Nash equilibrium to begin with.
There is, in any case, a further possible interpretation of Van Huyck, Battalio and Beil’s
experimental results, stemming from non self-enforcing Nash equilibrium, that may be more
compelling. Indeed, it may provide a more compelling case of non self-enforcing Nash
equilibria than does the Prisoner’s Dilemma itself. The outcome in which all players choose the
maximum effort, E, is a strict Nash equilibrium, and the unique Pareto optimal outcome, and
hence naturally focal. (Schelling, 1980) With the focal property, it is as if players do “start at,”
mentally, that optimal Nash equilibrium. But that equilibrium is never observed in the
experiments. Notice that effort above the minimum effort level of the players is wasted, this is a
“weakest link” technology. Thus it seems that many players are “undercutting,” choosing a
lower effort level than E, out of their fear of someone else undercutting, out of his fear of
someone else undercutting, and so on. In other words, accepting the focal argument, in this
weakest link environment Nash equilibrium is not self-enforcing, and the outcome not
determinate as a consequence.
Further, behavior in their repeated experiments exhibited a downward cascade. In
their experimental form Van Huyck, Battalio and Beil used seven levels of “effort,” and payoffs
were in dollars. In the first experiment there were seven groups of 14-16 subjects playing ten
repeated games with the same group. In the first play almost 70% did not play the high, best for
all, if all do so, effort level, but only 2% played the lowest effort level. Only 10% predicted there
would be an equilibrium outcome. In subsequent repetitions, many of the players played below
the minimum effort level of the previous repetition, that is learning, or at least reaction to
previous play, was observed. Perhaps the preceding minimum effort level became focal for
some, but not self-enforcing due to the fear of undercutting. By the tenth period 72% of the
subjects played the minimum effort level, but a Nash equilibrium was never observed. Thus
while common expectation, or “common knowledge,” was not achieved in this interaction,
greater concentration in behavior did develop. It is worth noting that while this “min rule”
specification imposes more severe tests of self-enforcement of Nash equilibrium as the number
of players grows, qualitatively similar results can even be generated in 2x2 versions of the
experimental game, and “To summarize, it is not appropriate to assume that behavior will
somehow converge to the best equilibrium for all, even when it is a Nash equilibrium” (Holt, c.
2007, p. 41).
Given the downward cascade of effort levels particularly, such models do seem to have the
potential for contributing to our understanding of instability. At the same time, that the
Prisoner’s Dilemma and the “min rule” may provide examples of non self-enforcing Nash
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equilibria does not rule out Nash equilibria typically being self-enforcing.
Generalizations of Proposition I provide further support to Nash Equilibrium as a solution
concept.
Proposition III: Suppose that the assumptions of the basic strategic game do somehow imply
some single outcome for every such game that has a pure strategy Nash equilibrium. Then that
outcome is a pure strategy Nash equilibrium.
Proof: The assumptions of the basic strategic game include that the players are rational
maximizers choosing freely among their strategies and with full information of the structure of
the game. If the solution were not a Nash equilibrium then at least one of the players could
increase his payoff, if the others did play their purported solution strategies, by deviating. We
cannot rule out such deviation by a rational maximizer choosing freely among his strategies and
with full information of the structure of the game. (Nor then can we in general rule out other
players deviating.)
Originally it may have been hoped that the assumptions of the basic strategic game would
generate a solution in the sense of predictions of particular unique outcomes. If so, pending
figuring out how it does so, Nash equilibrium provides all the possibilities. The approach that
behavior in simple choice settings is sufficient for determining social behavior has been a very
fruitful one in Economics, Walrasian equilibrium being the lead example. So such hope for the
basic strategic game would have been natural.
Proposition IV: Suppose the assumptions of a particular basic strategic game, that has a pure
strategy Nash equilibrium, can be augmented with assumptions on the environment that leave the
strategy spaces and payoffs unchanged, and the players rational maximizers choosing freely
among their strategies and with full information of the structure of the game. If this augmented
game implies a single outcome it is a pure strategy Nash equilibrium of the original game.
Proof: (See the proof of Proposition III)
Paraphrasing, Proposition IV says that, under the right conditions, if there is a solution, in
the sense of an implied single outcome, it is a Nash equilibrium.
Notice further that if the Nash equilibria in a game are not self-enforcing then under the
conditions of Propositions III and IV there can be no single outcome that is implied. For then, by
definition, deviation from a Nash equilibrium cannot be ruled out.
Proposition IV’s assumptions violate the complete description assumption, but they do
preserve the property that Nash equilibria provide all the possibilities for solution, given that
there is a solution concept as characterized in the Proposition. Of course this Proposition leaves
open what, and how situation specific, the additional assumptions are, if there exist such
assumptions. It is worth keeping in mind again that in a real world situation, of the sort being
modeled with strategic games, just one outcome happens, in a single play anyway, and,
presumably, something makes this so. The question is the nature of the something, namely
whether it meets the Proposition IV assumptions. Pending figuring out the additional
assumptions on the environment, Nash equilibrium provides the possible candidates for the
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implied single outcome from additional elements of the environment meeting the conditions of
Proposition IV, if such exist.
Indeed, pending having a set of additional assumptions as characterized in Proposition IV
in hand, or, on the other side, pending a demonstration of their impossibility, Nash equilibrium
might well be thought of as a solution concept. Notice, however, that even if the original basic
strategic game has a unique pure strategy Nash equilibrium this uniqueness, of itself, does not
resolve the fundamental issue of why the players would play a Nash equilibrium. The additional
assumptions of Proposition IV are doing some heavy lifting, not only in determining which Nash
equilibrium, but also why equilibrium at all, and arguably should be a major component in the
modeling. Just using Nash equilibrium on a wing and a prayer is open to question if there is an
alternative. After all, the Proposition IV assumptions are purely hypothetical. Still, on the other
hand, the additional factors generating Nash equilibrium play may be very real, but hard to
specify.
Still, Proposition IV may provide the strongest case for Nash Equilibrium as a solution
concept for the basic strategic game, indeed possibly even an ideal solution concept involving
alterations or supplements to the assumptions of the basic strategic game and the resulting logical
implication for behavior being a unique outcome. Moreover, the basic strategic game does seem
to describe the strategic elements of at least some social interactions, else how would it be
possible to structure experiments on them, see e.g. Holt (2007)? Further, experiments may strip
away the Proposition IV additional environmental elements or features that generate a unique
outcome.
V. COMMON KNOWLEDGE
There are indeed approaches in the literature, implicitly somewhat in the vein of
Proposition IV, to solving the basic strategic game, as succinctly summarized by Aliprantis and
Chakrabarti (2011, pp. 48, 55) :
“What should be the notion of a ‘solution’ {for a matrix [two player strategic] game}? … Nash
equilibrium, a fundamental notion for game theory. … it is in the [unambiguous] interest of a
player to play a Nash equilibrium only if the player is quite certain that the others are going to
play the Nash equilibrium. Often it requires that each player know this, that every player know
that every player knows this, and so on ad infinitum. That is it must be common knowledge that
the players are going to play the Nash equilibrium. ... a Nash equilibrium strategy profile is
self-enforcing. Hence, if the players are searching for outcomes or solutions from which no
player will have an incentive to deviate, then the only strategy profiles that satisfy such a
requirement are the Nash equilibria.”
The necessity of common knowledge to justify Nash equilibrium is widely accepted as
Aliprantis and Chakrabarti point out. Common knowledge is assumed to imply that all the
players know how the others will play (Osborne, 2004, p. 21), which seems to go a long way
towards assuming away the heart of the problem. A critical question then is how the players
come to have common knowledge. Often common knowledge is justified by prior repeated
interaction among the players in the game. (Dixit and Skeath, c. 2004, p. 89, Osborne, 2004, p.
12
21) While common knowledge was not achieved in Van Huyck, Battalio and Beil’s famous
(1990) repeated experiments, for example, greater concentration in behavior did develop. This
leaves open the question of when and how repeated interaction, of what form, generates common
knowledge. For one thing, in some settings, players may have motive to try to give the others
misconceptions about their behavior prior to the game at hand. Also, prior to achieving common
knowledge, players presumably would infrequently encountered Nash equilibrium play. Further,
game theory based on the Nash equilibrium solution concept can only be justified in this manner
when such extensive player interaction has actually occurred. This approach, of course, violates
the complete description assumption, and the assumption of isolated players given that their prior
interaction affects behavior. Ideally, one might also have the explicit, rigorous modeling of the
interaction as a, perhaps the, critical component of a game theoretic model so motivated, that is a
dynamic model of learning perhaps. In any case, if common knowledge is achieved, the
outcome is a Nash equilibrium of the basic strategic game. Common knowledge of pure
strategies implies a single outcome, and by Proposition IV it is a pure strategy Nash equilibrium.
Just achieving common knowledge does not specify which Nash equilibrium if there are multiple
ones however. Also, whether this can be thought of as a solution to the basic strategic game is
another matter, given the violation of the isolated players assumption. Of course one might also
wonder, for example, whether such extensive and intensive interaction might not possibly result
in non-Nash tacit cooperation in the Prisoner’s Dilemma for example!
VI. AVOIDING DEVIATION
Another justification for Nash equilibrium as solution concept is the above Aliprantis and
Chakrabarti (2011, p. 55) “searching for outcomes or solutions from which no player will have
an incentive to deviate.” If referring to the strategic game itself, this “searching” must be a
metaphor for individual mental activity, done in isolation prior to play. Why do players do this,
particularly as a Nash equilibrium need not have good welfare properties? An immediate
reaction after all is that our players may eat bread, but they do not eat non-deviation. In the basic
strategic game what they are doing is trying to rationally maximize their payoff, and payoffs
depend only upon the outcomes. Yet a player supposedly is going to pick a strategy of some
Nash equilibrium on the chance that the others pick their strategy of that Nash equilibrium,
because no one would have an incentive to deviate from it if they knew the others would do it,
which they do not know, unless this argument implies that they do know, but it does not, for that
is circular reasoning: “if they all thought they all would do it, they all would do it, therefore they
all do it.” This deviation, too, is not a dynamic event, but a thought experiment. And a thought
experiment that does not answer the observation that if a player’s play did, in fact, turn out to be
a deviation from play of a particular Nash equilibrium by the other players, and if the player had
known what was going to happen he would have acted differently, but he did not know what was
going to happen. Also, how do the players “converge” (so to speak) on the same Nash
equilibrium outcome when there are multiple Nash equilibria? At least as described above, this
deviation avoidance mechanism might require further development to reach the high bar of
producing common knowledge.
Perhaps a sketch of a simple environment that does exhibit deviation avoidance of a sort
will help illustrate the challenges associated with modeling deviation avoidance. One of the
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players in an, otherwise, basic strategic game has a technology, a magic wand, for making any
one of the outcomes focal (which violates the isolated players assumption). (Myerson, 1991, p.
111) That is to say, unless there is a reason for some player(s) individually and alone not to do
so, each player will assume that the others will play their strategy of the focal outcome, and
consequently do so themselves. Thus the magic wand insures that, if chosen, a Nash equilibrium
is self-enforcing. Then, if it is a good outcome for him, the player with the focal technology will
choose the Nash equilibrium in which his payoff is highest. Notice that this mechanism resolves
both which equilibrium and why equilibrium at all, if the focal arbitrator acts. Perhaps this
sketch is enough to suggest that further development is both necessary and worthwhile to
produce a rigorous Proposition IV solution concept in this deviation avoidance vein.
Admittedly, the focal approach for determining which equilibrium, and for resolving the
more basic question of why to play equilibrium strategies to begin with, does not require the
device of a focal arbitrator. In some games, like Bryant’s “min rule,” the very structure of the
game itself determines a focal outcome, but in many strategic interactions the focal could depend
upon aspects of the environment outside the strategic structure, as emphasized by Schelling
(1980). Still, elaborations of the focal arbitrator could provide interesting insights into the
purposeful generation of the focal.
VII. VON NEUMANN AND MORGENSTERN CARDINAL UTILITY AND THE MIXED
STRATEGY NASH EQUILIBRIUM SOLUTION CONCEPT
So far the issue of games without pure strategy Nash equilibria has been skirted, and
there most certainly are such games. For example, nine of the seventy-eight completely ordered
2x2 ordinal games do not have pure strategy Nash equilibria. The standard approach to these
games is to assume Von Neumann Morgenstern [vNM] cardinal utility, not ordinal utility, and
treat probability distributions over the pure strategies, that is treat mixed strategies. Cardinal
utility and probabilities together enable expected utility analysis, and the ranking of mixed
strategies thereby. To an extent, this works on games without pure strategy Nash equilibria,
consider Nash’s famous theorem:
“Proposition 119.1 (Existence of mixed strategy Nash equilibrium in finite games) Every
strategic game with vNM preferences in which each player has finitely actions has a [at least
one] mixed strategy Nash equilibrium.” (Osborne, 2004, p. 119)
In particular, “every” includes games without pure strategy Nash equilibria. A mixed
strategy equilibrium meets Myerson’s (1991, pp. 88) goal of enabling one to “assess the
probability of the various strategies,” in that equilibrium anyway.
VIII. STABILITY
Yet this mixed strategy approach to the problem of no pure strategy Nash equilibria only
works to an extent:
14
“… the [non-degenerate] mixed-strategy equilibria are Nash equilibria only in a weak sense [that
is, they are non-strict].
This property seems to undermine the basis for mixed-strategy Nash equilibria as the
solution concept for games. … Why not just do the simpler thing [than mixing] by choosing one
of his pure strategies [in the mixture]? After all, the expected payoff is the same. The answer is
that to do so would not be a Nash equilibrium; it would not be a stable outcome, because then the
other player would not choose to use his mixture.” (Dixit and Skeath, 2004, p. 244)
That the mixed strategy Nash equilibria are necessarily weak does seem to weaken the
case for them being a solution concept for the basic strategic game. Deviation from a nondegenerate mixed strategy equilibrium is not costly and cannot be ruled out in the static setting of
the basic strategic game, as the players are rational maximizers choosing freely among their
strategies and with full information of the structure of the game. In brief, these equilibria are not
self-enforcing in a static setting, and, hence, direct analogues to Propositions III and IV are not
available.
Dixit and Skeath suggest that players play mixed strategy Nash equilibrium strategies,
which gains them nothing in the immediate sense, because otherwise they would not be playing a
Nash equilibrium. For “then the other player would not choose to use his mixture” and it is “not
a stable outcome.” These suggested reasons seem to demand a dynamic game setting in which
current play affects future play. Ideally, then, one might have the explicit, rigorous modeling of
the setting of the dynamic game generating the equilibrium play as a critical component of the
mixed strategy Nash equilibrium model so motivated. It would also seem that such support of
mixed strategy Nash equilibrium as solution concept might be valid only in the context of such a
dynamic game setting.
IX. CONCLUSION
Iterated elimination of set and vector dominated strategies are derived from explicit
extensions of maximization in simple choice settings to strategic interaction, as suggested by Von
Neumann and Morgenstern (c. 1944, Chapter I, 2. -2.2.4). Thereby they provide the simplest
solution concepts for basic strategic games, but they often only imply modest restrictions on
strategy choice. Consequently the solution concept of choice for such games has been Nash
equilibrium, which typically further restricts behavior. Nash equilibrium not evolving from such
an extension of maximization to strategic interaction, there is a tradeoff in the use of Nash
equilibrium as a solution concept however. Specifically, it requires that the players have
common expectations without the structure of the game itself necessarily providing a mechanism
for this occurring. Consequently Nash equilibrium really seems to require further support for its
use as solution concept.
A promising approach to the support of the pure strategy Nash equilibrium as solution
concept is the possibility for supplements to the assumptions of the basic strategic game that
yield the prediction of a single outcome. If these new assumptions leave the strategy spaces and
payoffs unchanged, and do not impede the players being rational maximizers choosing freely and
with full information of the structure of the game, as in the basic strategic game itself, then that
predicted outcome is a pure strategy Nash equilibrium of the original game. Further, prominent
discussions in the literature supportive of pure strategy Nash equilibrium as solution concept, and
15
implicitly in this vein of new assumptions for the basic strategic game, are promising and
warrant further attention and development. In particular, these can suggest dynamic
environments that may be worth explicitly including as an integral component of the game
theoretic models themselves. Finally, mixed strategy Nash equilibria are weak, and thus not selfenforcing, and may consequently be of a more provisional character as a solution concept. Still,
mixed strategy Nash equilibrium may be supported in a dynamic game setting, which too might
fruitfully be explicitly included as an integral component of the models. This approach may
improve our understanding of environments that are conducive to equilibrium play, particularly
important given the extensive role for Nash equilibrium in game theory, and for game theory
itself, that game theorists advocate for them.
At the same time, it could just be, for example, that the basic strategic game does capture
all the relevant strategic elements, but there are non-strategic elements at work helping determine
the outcome, and these non-strategic elements are hard to identify and specify. If these nonstrategic elements also leave the strategy spaces and payoffs unchanged, and do not impede the
players being rational maximizers choosing freely and with full information of the structure of
the game, then pure strategy Nash equilibrium provides all the candidates for solution, in the
sense of predictions of particular unique outcomes. This should suffice for “solution concept”
status.
“To know game theory is to change your lifetime way of thinking. [Dixit and Skeath] is a
delightful skeleton key to the twenty-first century’s emerging culture.” (Paul A. Samuelson,
Dixit and Skeath, c. 2004, back cover)
RICE UNIVERSITY
16
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FIGURE I
Prisoner’s Dilemma
19