Consider the following two player game, where player 1 chooses rows, and player
ID: 3317216 • Letter: C
Question
Consider the following two player game, where player 1 chooses rows, and player 2 chooses columns Game L U (2,0 1,-1) MD (0,0 -1,0) Imagine that instead of the payoffs we used an increasing linear transformation of those payoffs. Would the set of Pure action Nash Equilibria change? would the set of mixed action nash equilibria change? Would the set of rationalizable actions change? e Imagine that instead of the payoffs we added a constant c > 2 two and then applied In. For example, instead of (MD, R) generating payoffs -1,0) we now make (MU, R) generate payoff (In(1c), In(0+c)). Would the set of Pure action Nash Equilibria change? would the set of mixed action nash equilibria change? Would the set of rationalizable actions change? . For the game represented in the matrix (i.e. disregard the transformations men tioned in the two items above) calculate the set of pure action Nash equilibria and the set of rationalizable actions Assume that each player knows his own utility function, and the utility function of his rival. Assume also that player 1 knows that he is rational and that player 2 is rational. Assume that player 2 knows that he is rational. Using only these assumptions, what actions do you predict players might takeExplanation / Answer
If increasing linear transformation is done, then Pure action Nash equilibria will not change. The set of Mixed action Nash equilibria will also not change. The set of rationalizable actions will also not change. This is because the payoffs will remain the same. If a constant c>2 is added to the payoffs and ln transformation is done, then Pure action Nash equilibria will not change. The set of Mixed action Nash equilibria will also not change. The set of rationalizable actions will also not change. This is because the payoffs will remain the same. The equilibria: The Pure strategy Nash equilibria are: (MU, R) = (1,1) and (D, L) = (2,2). The Rationalizable actions are: (MU, R) = (1,1) and (D, L) = (2,2). If player 1 knows his rational and that of player 2 and player 2 knows his own rational then the equilibria will be (MU, R) = (1,1) and (D, L) = (2,2). it is beacuase although player 2 will have (D, R) as payoff but it is not rational for player 1; so player 1 will never choose it. If the players know each others strategies then the will reach a sub-optimal equilibrium (MD, L) = (0,0) as they will try to maximise their own payoffs but ultimately reaching a sub optimal position. The SPNE strategy profile is: (D,L) -> (MU,L) -> (D,L) Since player will play (D, L) for first 9 periods, then at the 10th, player 2's best output is MU & then from the two, L is the best for player 1.