Strategy: Rule or plan of action for playing the game.
Game: A situation where players make strategic decisions,
decisions which take into account each others actions and responses.
Payoffs: Results of strategic decisions. They are outcomes
which generate rewards or benefits. For example for a price setting firm
payoffs are profits.
Optimal strategy: The strategy which maximizes a player’s expected
payoff.
Dominant strategy: The strategy which is best for me regardless of what
my opponent does.
DOMINANT AND NASH EQUILIBRIUM
Let us consider 2 firms A and
B selling competing products and deciding whether or not to advertise. The
payoffs resulting from different strategies are given below.
Each firm
will be affected by its competitor’s decision.
Let us
consider firm A. it should clearly advertise because no matter what firm B does
A does best by advertising. If B advertises A earns a profit of 10 if it
advertises but only 6 if it doesn’t. If B does not advertise A earns a profit
of 15 if it advertises but only 10 if it does not. Thus advertising is a
dominant strategy for A.
The same
is true for B. no matter what A does B does best by advertising. Therefore
assuming both firms are rational the outcome of the game is that both firms
will advertise. When each player has a dominant strategy we call the outcome of
the game equilibrium in dominant strategies.
Now let
us consider a different pay off matrix.
Now firm
A has no dominant strategy. Its optimal
decision depends on what B does. If B doesn’t advertise A also does best by not
advertising.
Now
suppose both firms have to take their decision at the same time. What will A
do? A will have to put itself in B’s shoes and think what B is most likely to
do. The answer is clear. B has a dominant strategy- to advertise no matter what
A does. Therefore firm A concludes that B will advertise. Therefore A should
also advertise and earn 10 instead of 6. The outcome of the game is that both
firms advertise because A is doing the best it can given B’s decision and B is
doing the best it can given A’ decision. This is a case of nash equilibrium
Dominant strategy
• A is doing the best regardless of
what B is doing
• B is doing the best regardless of what
A is doing.
Nash equilibrium
• A is doing the best given what B is
doing
• B is doing the best given what A is
doing
MAXIMIN STRATEGY
The
concept of nash equilibrium depends a lot not just on the rationality of the
player but also of the opponent. This can often be a limitation.
Let us
consider 2 firms considering whether or not to invest
Rational:
Here investing is a dominant
strategy for firm 2 because in doing so it will do better regardless of what 1
does. Here 1 will also do better by investing and earning 20 crore than by not
investing and losing 10 crore. Therefore (invest, invest) is the nash
equilibrium. Here we assume that firm 1 is sure that firm 2 understands the
game and are rational.
Not rational:
Suppose firm 1 is cautious
and not sure whether firm 2 is fully informed and rational. They are not sure
whether 2 will choose to invest or not. Thus 1 might choose to not invest. In
this case the worst that can happen is that it will lose 10 crore. But that is
still better than losing 100 crore which is what would have happened had 2 not
invested and 1 had. This strategy is called a maximin strategy because it
maximizes the minimum gain that can be earned. If both firms used maximin
strategy the outcome would be that 1 does not invest and 2 does. A maximin
strategy is conservative but not profit maximizing.
PRISONERS DILEMMA
This is one of the most
popular examples to explain game theory. The numbers indicate the years of
imprisonment.
Thus (confess, confess) is a
dominant strategy as well as a nash equilibrium.
-Noyonika Bose
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