Question

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Consider the following game: Player B Left Right Up 4,1 0,0 Player A Down 0,0 1,4 a) Find all Nash equilibria in this simultaneous game (including the mixed strategy equilibrium) and illustrate them in a graph showing the best response functions. (12p) b) Now assume that player A can choose his/her action before player B and that player B can observe what player A has chosen before B makes his/her choice. Draw the sequential game in extensive form and find the equilibrium (subgame perfect equilibrium). Is there a first mover advantage in this game? (8p) c) Write down the proper normal form (two by four) of the sequential game in b) and find all Nash equilibria in pure strategies. Explain the relationship between these equilibria and the subgame perfect equilibrium. (5p)

Answer 1

solution: Given Game is, player B left Right Up 4,13 0,0 Player A down 9,0 1,40 @ all nash equilibria are: upcu), then player B will choose - If player a chooses L of 130) • Similarly coher player o choose L, player Awill choose opcu) of (uso). ;. So (U/L) is a pure stategy nosh equilibrium > If player A chooses Down (D), then player B will choose Right (R) of (uso) Similarly if player B chooses R then playes I will

choose down af (120) equilibrium so (B, R) is a pure strategy nash - for mixed strategy, let playes A chooses u with Probability pand D with probability Clap) and player B chooses L with probability q and R with the probability (1-9) So EUACU) = 49 +0 (19) = 49 EUA.CO) = 0 or +1(1-9) =129 So, 49=1-9 or 5931: q = 1/5 and 1-9 = 7115 =915 EUB(1) = 1 (P) + 0(1-P) = p. EUB(K) = O(P) + 4C1-P) = 404p

So, p=4-up or sp=4 p=415 and 1-p = 1-415 = 1/5 So if EU ACUJ Z EUA (D) then player I will choose u with pal. 1=). || 42 ام و Similarly when, EUBCL) Z EUB (R) =) P2415 then player B will choose L so q=1. qli ०१७ (b) If player A can choose his her action before player B and then

Lul Player A At node I player B chooses as 120, and at node (2) bol u zo Player B chooses R as 420 ul so player it I in. So at final node playes A chooses vas (451) So subgame perfect equilibrium = { up, Left Right} : Yes player A gets first mover advantage all nash equilibrium © using a proper normal form proper normal puse strategies are The values are:

LLILR RL 0,0 1,4 when playes A chooses o, playes B will choose LL and/ LR as (170) then, player a Similarly when player to chooses LL will choose u of (4>0) / and when player B chooses LR, player A will choose So (O, LL) and Co, LR) are pure strategy hash equilibria 1. Similarly (D, RR) also puste strategy nash equilibria This is a relationship between Nash perfect equilibria. equilibria and subgame % Subgame perfect eq = (ULR) , (L) can never be reached of ll is non because credible threat