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Game Theory with Applications to Finance and Marketing

Game Theory with Applications to Finance and Marketing

simultaneously at each

simultaneously at each stage. Is the game in example 3 dominancesolvable? 1 (Hint: Player 1 is not indifferent about (M,R) and (M,L).)10. Consider the following two-player game. They each announce a naturalnumber a i not exceeding 100 at the same time. If a 1 + a 2 ≤ 100, thenplayer i gets a i ; if a 1 + a 2 > 100 with a i < a j , then players i and j getrespectively a i and 100 − a i ; and if a 1 + a 2 > 100 with a i = a j , theneach player gets 50. Determine if this game is dominance solvable.11. Consider example 1 with the further specifications that c = F = 0,P (q 1 + q 2 ) = 1 − q 1 − q 2 . Like the game of prisoner’s dilemma, there isa unique outcome surviving from the iterated deletion of strictly dominatedstrategies. We say that this game has a dominance equilibrium.To see this, note that the best response for firm i as a function of q j(called firm i’s reaction function) isq i = r i (q j ) = 1 − q j.2Note that q 1 > 1−0 = r2 1 (0) is strictly dominated for firm 1. 2 Sincerationality of firm 1 is the two firms’ common knowledge, firm 2 will realizethat any q 2 < 1− 1 2= r2 2 (r 1 (0)) is strictly dominated by r 2 (r 1 (0)). 31 Definition 5 can be found in Osborne and Rubinstein’s Game Theory. A relateddefinition is the following: A game is solvable by iterated strict dominance, if the iterateddeletion of strictly dominated strategies leads eventually to a unique undominated strategyprofile; see Fudenberg and Tirole’s Game Theory, Definition 2.2.2 Observe that (i) given any q j ∈ R + , Π i (·, q j ) is a strictly decreasing function on[r i (q j ), +∞); and (ii) r i (·) is a strictly decreasing function on R + . Thus if q i > 1−02= r i (0),then for all q j ∈ R + , we haveq i > r i (0) ≥ r i (q j ) ⇒ Π i (r i (0), q j ) > Π i (q i , q j ),proving that any such q i is strictly dominated by r i (0), regardless of firm j’s choice q j .3 Note that Π 2 (q 1 , ·) is a strictly increasing function on [0, r 2 (q 1 )]. Since by commonknowledge about firm 1’s rationality, firm 2 believes that any q 1 chosen by firm 1 mustsatisfy q 1 ≤ r 1 (0), we must haveq 2 < r 2 (r 1 (0)) ≤ r 2 (q 1 ),and henceΠ 2 (q 2 , q 1 ) < Π 2 (r 2 (r 1 (0)), q 1 ),6

But then, common knowledge about rationality again implies that anyq 1 > 1− 1 4is strictly dominated for firm 1, which in turn implies that2q 2 < 1− 3 8is strictly dominated for firm 2. Repeating the above argument,we conclude that firm 1 will find any q 1 > 1− 5162strictly dominated.Thus we obtain a sequence q1 n = 4n −(1+4+4 2 +···+4 n−1 )2of undominatedoutput levels at the n-th stage of deleting strictly dominated2×4 nstrategies. Since the sequence converges uniquely to 1 , we conclude3that the only output level surviving from the iterated dominance processis 1 for both firms. Thus, 3 (q∗ 1, q2) ∗ = ( 1, 1 ) is the (unique) dominance3 3equilibrium.12. (Market Share Competition)player 1/player 2 Don’t Promote PromoteDon’t Promote 1,1 0,2-cPromote 2-c,0 1-c,1-cIn the above, two firms can spend c > 0 on promotion. One firm wouldcapture the entire market by spending c if the other firm does not doso. Show that when c is sufficiently small, this game has a dominanceequilibrium.13. Consider also the following moral-hazard-in-team problem: Two workerscan either work (s=1) or shirk (s=0). They share the output4(s 1 + s 2 ) equally. Working however incurs a private cost of 3. Showthat this game has a dominance equilibrium.14. Definition 6: A pure strategy Nash equilibrium (NE) for a game innormal form is a set of pure strategies (called a pure strategy profilefrom now on), one for each player, such that if other players play theirspecified strategies, a player also finds it optimal to play his specifiedstrategy. In this case, these strategies are equilibrium strategies. Forinstance, (U,R) and (D,L) are two pure strategy NEs for the gameproving that any such q 2 is strictly dominated by r 2 (r 1 (0)) from firm 2’s perspective, nomatter which q 1 chosen by the rational firm 1.7

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