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Game Theory with Applications to Finance and Marketing

Game Theory with Applications to Finance and Marketing

u i (σ i , σ −i )

u i (σ i , σ −i ) for all σ i ∈ Σ i ). A (mixed) strategy σ i is weakly dominatedby σ ′ i for player i if for some σ′ i ∈ Σ i,u i (σ i , σ −i ) ≤ u i (σ ′ i , σ −i), ∀σ −i ∈ Σ −i ,and σ i is strictly dominated if the above inequalities are all strict. Ourfirst equilibrium concept is that rational players will not use strictlydominated strategies. Consider the following normal form game, knownas the prisoner’s dilemma:player 1/player 2 Don’t Confess ConfessDon’t Confess 0,0 -3,1Confess 1,-3 -2,-2In this game, “Don’t Confess” is strictly dominated by “Confess,” andhence the unique undominated outcome is the one where both playersconfess the crime. Note that unlike Walrasian equilibrium, this equilibriumis Pareto inefficient. That equilibria in a game are generallyinefficient is the first lesson to be learned here (and all kinds of economictheory starts from here).6. Based on the implicit assumption that the whole normal form game isthe players’ common knowledge, the above dominance argument can beextended further so that we shall be looking at outcomes that survivefrom the procedure of iterative deletion of strictly dominated strategies.The following is an example.player 1/player 2 L M RU 0,-1 0,0 1,13M 2,3 3,1 2D 4,2 1,1 2, 3 2Note that M is not strictly dominated by L from player 2’s perspective,but since U is strictly dominated by M from player 1’s perspective,and in the absence of U, M is strictly dominated by L from player 2’sperspective, we should not expect player 2 to use M. It follows thatplayer 2 will use L and hence player 1 will use D.4

Observe that we have repeatedly used the assumption that rationalityof all players is the players’ common knowledge. For example, if player2 is not sure about player 1’s being rational, then player 2 may notwant to ignore strategy M.7. Mixed strategies that assign strictly positive probabilities to strictlydominated pure strategies are themselves strictly dominated. A mixedstrategy that assigns strictly positive probabilities only to pure strategieswhich are not even weakly dominated may still be strictly dominated.Consider also the following example:player 1/player 2 L RU 1,3 -2,0M -2,0 1,3D 0,1 0,1The mixed strategy (0.5, 0.5, 0) for player 1 is strictly dominated by D.8. It can be shown that iterated deletion of strictly dominated strategieswill lead to the set of surviving outcomes which is independent of theorder of deletion. The same is not true for iterated deletion of weaklydominated strategies. Consider the following normal form game:Example 3:player 1/player 2 L RU 1,1 0,0M 1,1 2,1D 0,0 2,1If we delete U and then L, then we conclude that the payoff profilewould be (2, 1). If we delete D and then R, then we conclude that thepayoff profile would be (1, 1).9. Definition 5: A normal form game is dominance solvable if all playersare indifferent between all outcomes that survive the iterative procedurewhere all the weakly dominated actions of each player are eliminated5

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