Question

13. Consider the following n-player game. Simultaneously and independently, the players each select either X, Y, or Z. The payoffs are defined as follows. Each player who selects X obtains a payoff equal to y, where y is the num- ber of players who select Z. Each player who selects Y obtains a payoff of 2a, where a is the number of players who select X. Each player who selects Z obtains a payoff of 3B, where ß is the number of players who select Y. Note that a + B + y = n. (a) Suppose n = 2. Represent this game in the normal form by drawing the appropriate matrix. (b) In the case of n = 2, does this game have a Nash equilibrium? If so, describe it. (c) Suppose n = 11. Does this game have a Nash equilibrium? If so, describe an equilibrium and explain how many Nash equilibria there are.

Answer 1

N- Player game choice = $x, 4, 2} x fay off = 9: gi number of people who selected Y Pay off = 2a; a'r number of people who selected & 2, Poy off = 366 bt number of people who seleclety atbigan at gbo - n=& choice = { x,y,z} له له 12 Coel 100 (13,0) 2 1 10 10,3) 110,9 1. b, n=2 the game in deed have no Neash equilibauum , Acash'equilibrium is that faredo distribution, when one Comunist gain.Soy unli beral change of skuborgy of other siemai inchanged.

Nach equilibrium is a Concept of game thely, where optimal outcome of a game is one where no flayer has an incenture to daruiale from his closen stategy after considering opercute choices. In the Care flayer always have a choice that will yeild him a hight pay off given other player staturgy only equilibrium pay . do tab gone has no Maks egallrisma. q n=11 now Pay offs are in faine 9, 2a, sh a, 1, 2, s. multiple now Lem of there a number will be 6. as = 9= 2a=6 a=3 36=6 b=2 g+ath=6+3+2 = 11=n So there will be anach equililoricense g=6, b=2, a=3 6 people select z 2 people select y § People Select y the number of wayls of the selection = "C6 Sez 36₂ - 10X1X462 = 4620