Equilibrium strategy

  (10%). Find and write down all Nash equilibria in the following game (payoffs on the left correspond to Player 1 and payoffs in the right correspond to Player 2): Player 1,Player 2 E F G A 5,6 3,7 0,4 B 8,3 3,1 5,2 C 7,5 4,4 5,6 D 3,5 7,5 3,3 2 (20%). Player 1 faces two options: A or B. If it chooses A, it obtains a payoff of 2.5. If it chooses B, player 1 will play the following game in pure strategies with player 2 (payoffs on the left correspond to Player 1 and payoffs in the right correspond to Player 2): Player 1,Player 2 L R U 3,4 1,1 D 2,3 2,2 With this information in hand, what is Player’s 1 best strategy: A or B and why? 3. Consider 2 identical players (i.e. i = 1, 2) with utility function: πi = b(qi + q-i) - cqi. Where qi is equal to one if player i contributes to the provision of a public good and zero if she does not, q-i is the sum of the contributions by all other players, b is the constant marginal benefit of contributing to the public good, and c is the cost of contributing to the good. Assume that c = 1. The players have two possible actions: to cooperate to the public good (C), or not to cooperate to the public good (NC). The players choose actions simultaneously and only one time. a) Write down the game in strategic form (20%). b) Suppose that player 2 will always play C (ie, will always cooperate). For what values of “b” will player 1 play C? Show your calculations (20%). 4 (30%). Jesse (2009) considers an ideal point model where the probability of a ‘yea’ vote by actor I on proposal j is p(yij=1|γ, α, x)=Φ(γj xi- αj), where xi is a voter’s ideal point and αj and γj are vote-specific difficulty and discrimination parameters. Here, Φ(.) is the cumulative density function (CDF) of the normal distribution, with associated probability density function (PDF) ϕ(.). Jesse extends this ideal point model to consider the effect of political information where a factor φi is a function of a voter’s information, denoted PolInfoi. In lecture we explored the effect of a change in a voter’s ideal point on the probability of a ‘yea’ vote; we mentioned that this effect is also known as a marginal effect. Now assume that xi, αj, and γj are constants. Write down the marginal effect of PolInfoi on the extended ideal point model, also known as a heteroskedastic ideal point model.