Question

(i) Show that if both players in a matrix game have equalizing strategies, then these strategies are optimal.

(ii) Find a 2 x 2 matrix game in which one player has an equalizing strategy which is not optimal.

Answer 1

Ans (1) An Equalizing strategy is one where, irrespective of the opponent’s moves, the outcomes of one’s moves remain same. In other words, it may also be called the optimum move in case it delivers the required outcome irrespective of the opponent’s moves.

While dealing with Zero Sum Game, if a strategy is not dominant, the outcomes will remain same unless there is application of dominant strategy. Thus, an equalizing game will be effective only in cases where the payoffs remain same. There also an assumption which is applicable here; that the opponent believes in minimizing your payoffs. In short, in games where opponents are pitted against each other, Equalizing strategy will work. However, in case of Cooperation games, Equalizing Strategy will not work.

Example of Equalizing Games: In Auctions, there are two bidders. A Has a high Bid of 100 Dollars. B has high Bid of 90 Dollars. While both of them start bidding, they are not aware of the high points. In a game of perfect information, B would know in advance that he cannot win as A has higher capacity to pay. In case of imperfect information, A and B have to guess the moves and bid amount. If A does not move after 90 Dollars, he will win as the outcomes of the auction are not depending upon the moves of B at all. Thus, one can conclude that not making a move may result in an Equalizing game too.

Ans (2) Assume that two players have the following game:

Alice	Max
	Vanilla	Strawberry	Chocolate
Chocolate	(1, -1)	(-1,1)	(0,0)
Strawberry	(-1,1)	(0,0)	(1, -1)
Vanilla	(0,0)	(1, -1)	(-1,1)

Max and Alice have different preferences. The values in the box show the values derived by Alice and Max in consuming different ice cream flavours. (0,0) quadrants show equalizing game.