# Economics of strategies and information

1. A notion of dominance studied in class is weak dominance (recall that its defini- tion only requires weak inequalities). An alternative notion is strict dominance. Given a game Γ ≡ [N,(Σi)i∈N ,(ui)i∈N ], for each i ∈ N and all σi , σ0 i ∈ Σi , say that σi strictly dominates σ 0 i if for each σ−i ∈ Σ−i , ui(σi , σ−i) > ui(σ 0 i , σ−i); that a strategy is strictly dominated if some other strategy strictly dom- inates it; and that a strategy is strictly dominant if it strictly dominates every other strategy. With these definitions, we can define a process similar to iterated elimination of dominated strategies (IEDS), which eliminates only strictly dominated strategies in each round—let us call the process so defined iterated elimination of strictly dominated strategies (IESDS). (a) It is known that the order in which weakly dominated strategies are eliminated affects the IEDS solution. Show that the elimination order matters for the following game. 2 L R 1 T 2, 1 3, 1 C 2, 1 1, 2 B 1, 2 3, 1 (b) Solution concepts such as IEDS and IESDS are meaningful in that they serve as a prediction of a given game; e.g., one may argue that players are very likely to play an IEDS solution. Then are the predictions obtained by IEDS in part (a) convincing? Why or why not? (c) In contrast with IEDS, IESDS has the advantage that it does not rely on the order of elimination of strictly dominated strategies. Show that this is true for 2 × 2 games1 . (For this reason, IESDS is generally perceived as a more robust solution concept than IEDS). (d) IESDS is not without limitations, however. In each round of elimination, IESDS deletes only strictly dominated strategies and as compared to IEDS, the process can fail more easily to produce a single strategy profilein the end. Construct an example of a 2 × 2 game for which an IEDS solution exists but an IESDS solution does not. 2 × 2 game is a game involving two players, each with two strategies.

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