Download Document

Error-Awareness and Equilibria in Normal Form Games
Peter Kriss ’07
Swarthmore College, Department of Mathematics & Statistics
Abstract
Though previous explorations of equilibria in
game theory have incorporated the concept
of error-making, most do not consider the
possibility of anticipation of errors. Instead
of treating them as inherently unpredictable,
I allow the awareness of error-making to
directly affect a player's choice of strategy
before any errors actually occur. I explore
the consequences of allowing players to be
estimate their opponent's error rate and
incorporate this information into an
expected payoff function. I show that if both
players are aware of a high error rate of
their opponent, a new, stable, non-Nash
equilibrium can be achieved.
The General n × n Game Matrix
Any normal form game can, by definition, be
Background Information
Nash Equilibrium
The Nash Equilibrium (NE) is the equilibrium
concept in game theory. It is defined as a set of
strategies such that no player has an incentive to
make a unilateral change of strategy. That is,
each player is playing the best response to his
opponent’s choice of strategy.
E I [Si ]   aij p j
j1
One important point to note is that these
results depend only on the players'
estimations of their opponent's error rate. If
players are using error-aware best response,
the game can move between equilibria
without errors ever being made. Since a
high estimate of the opponent's error rate
can cause a player to change strategies,
there can exist an incentive for a player to
give the impression that his error rate is
higher than it is. If successful, this tactic
could force the opponent to play a higherror best response. Then, the deceptive
player might play a traditional best
response to this strategy and thereby arrive
at a higher payoff outcome.
E I[Si ]   ( p j (1II )  II /n)aij
j1
E II [T j ]   bijqi
That is, the payoff depends on the opponent’s
probability distribution.


The Error Rate
We now introduce the error rate, . Let  represent the
probability that a player's choice of strategy is not
executed as such, but a strategy is instead chosen
randomly (Young, 1998).
n
n
i1
Deceptive Error Rates.
Error-Adjusted Expected Payoff Functions
With errors incorporated into our expected payoff
functions, we find that the expected payoff to Player I
of playing Si and to Player II of playing Tj are now:
Expected Payoff Functions
As can be inferred from the n x n game matrix,
the expected payoff to Player I of playing Si and
to Player II of playing Tj are as follows:
n
Incorporating Errors

n
E II[T j ]   (qi (1I )  I /n)bij
i1
High-Error Best Response
In general, the strategies
that
maximize
these
functions

depend on the p’s and q’s. But as 1, they do not. For
some  and greater, a single strategy will always be the
best response. The set of these High-Error Best
Responses is the High-Error Equilibrium (HEE).
An Example: The Big Risk Game
We have three Nash Equilibria: (A,A), (B,B) and (C,C). Why?
represented as a matrix like the one below.
High-Error Best Response
The a- and b-values, S’s and T’s, and q’s and
Imagine we start at (A,A). Now if Player II’s error rate
increases past the threshold, Player I will switch to
strategy B. (Why?) Then, if Player I’s error rate increases
past the threshold, Player II will switch to strategy C, the
High-Error Best Response.
p’s are the payoffs, strategies, and
probabilities of playing those strategies of
Players I and II, respectively.
What do these two situations
have in common?
High-Error Equilibrium
So at (B,C), we have the High-Error Equilibrium. But this
High-Error Equilibrium not a Nash equilibrium of the
original game! What happened to NE being the
equilibrium concept?
Nash vs. High-Error Equilibria
If we reexamine the Nash equilibrium concept, the fact that the HEE is not necessarily a NE will not surprise us.
One formulation of the Nash equilibrium concept is a set of strategies such that there is no incentive for unilateral
deviation. But for significant error rates, the players have good reason to suspect that a deviation would not be
unilateral -- it would be bilateral. Thus, this error-making environment takes us out of the traditional Nash
equilibrium context.
Reference
Young, H. Peyton. 1998. Individual Strategy and Social Structure: An
Evolutionary Theory of Institutions. Princeton University Press,
Princeton.
Acknowledgements
I would like to thank Dr. Robert Muncaster of the University of Illinois
without whose introduction and guidance in evolutionary game theory,
this project would not have been possible.