ECON30290
MATHEMATICAL ECONOMICS IIA
Question 1. (a) Define and interpret the notion of a correlated strategy in a two-player two-strategy game.
(b) For the game
describe the set of all payo§ allocations (u(γ); v(γ)), where γ ranges over the set of all correlated strategies (u and v denote the payo§ functions of players 1 and 2, respectively). Draw a diagram depicting the set of all such payo§ allocations represented by points in the plane.
(c) Define the notion of an implementable payoff allocation. Explain what an individual rationality constraint is. Find the set of all imple- mentable payoff allocations for the game specified in (b). By using the diagram constructed in (b) show what points in the plane belong to the set of implementable payoff allocations. [20 marks]
Question 2. Consider the game:
(a) Find best responses p = BR1(q) of player 1 to mixed strategies q of player 2 and best responses q = BR2(p) of player 2 to mixed strategies p of player 1. Draw a diagram depicting the graphs ofthe best response mappings (reaction curves) in the plane. By using this diagram find pure strategy and mixed strategy Nash equilibria in the game under consideration.
(b) By using the diagram constructed in (a), find out what will be the limiting behaviour (as t → ∞) of the fictitious play dynamics in the following three cases: if the fictitious play starts from the initial state (p(0); q(0)) = (0.2; 0.4), if it starts from (p(0); q(0)) = (0.5; 0.8), and if it starts from (p(0); q(0)) = (0.9; 0.8). Draw the trajectories of fictitious play starting from the above initial states. [20 marks]
Question 3. (a) Let G be a two player game with strategy sets A and B and payo§ functions u(a; b) and v(a; b). Consider the infinitely repeated game with the stage game G and a discount factor 0 < δ < 1. Suppose that at each stage t = 0; 1; 2; ..., player 1 makes a move at and player 2 simul- taneously makes a move bt. What is the history of the infinitely repeated game and what are the playersípayoffs corresponding to this history? What is a strategy of a player? What is the outcome of the game given strategies of players 1 and 2? When do we say that a pair of strategies of players 1 and 2 forms a Nash equilibrium in the infinitely repeated game?
(b) Consider the Bertrand duopoly game with two firms 1 and 2 simulta- neously setting prices p1 and p2 for their homogeneous output whose demand function is D(p) = 1 - p. Sales for firm i are given by
(j ≠ i). The firms' profits are
πi(p1; p2) = (pi - c)Di(p1; p2); i = 1; 2;
where 0 < c < 1.
Let pM be the monopoly price, that is, the price maximizing (p -c)D(p) = (p-c)(1-p). Clearly pM = (1+c)=2, which yields monopoly profits π M = (1-c)2 =4. It is known that the only Nash equilibrium in the above Bertrand game is p1 = p2 = c, yielding zero profits. Show, however, that in the infinitely repeated Bertrand game with a "not too small" discount factor 0 < δ < 1, there might be a symmetric Nash equilibrium in which, at every stage, the outcome is p1 = p2 = pM , each firm earning π M =2 > 0 profits each period.
Consider the following trigger strategy σ* :
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Set the monopoly price p = pM in the first period. In the t th period, charge the price pM if both firms have charged pM in each of the previous periods; otherwise, set p = c.
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Suppose that the discount factor δ in the infinitely repeated game at hand is as follows:
where d is the fourth digit in your student ID. (For example, if your ID is 10154977, then d = 5.)
Does the strategy σ * form a symmetric Nash equilibrium in the infinitely repeated Bertrand game? Justify your answer. (If your answer is positive, you have to prove that σ * is a best response to itself; if it is negative, you have to show that σ * is not a best response to itself.) [25 marks]
Question 4. There are N shops on a road of length 1 represented by the segment [0; 1] in the real line. The shops sell a homogeneous good, and customers (uniformly distributed on [0; 1]) select the closest shop. If x1 ; ...; xN are the locations of the shops, then the payoff/profit of shop i is proportional to half of the length of the segment [x k; xj] where xk and xj are the closest neighbours of xi. We will assume that the coefficient of proportionality is equal to 1, so that i's payoff is exactly equal to half of the length of this segment:
If there is only one neighbour of xi, say xj on the right, then iís payoff is xi + (xj - xi)=2:
If there is only one neighbour x k on the left, then iís payoff is (1 - xi) + (xi - x k)=2:
If several shops occupy the same position, they divide the payoff evenly. Thus we have an N-player game with strategies xi ∈ [0; 1] specifying the shops' locations and the payoff functions (profits) defined above.
Prove that in the case of four players (N = 4) the strategy profile x1(*) =
x2(*) = 1=4, x3(*) = x4(*) = 3=4 (shops 1 and 2 are located at the point 1=4 and
shops 3 and 4 at the point 3=4) forms a Nash equilibrium in the game at hand. Prove the following uniqueness result: if (x1(*); x2(*); x3(*); x4(*)) is a Nash equilibrium in the game at hand, then two of the points xi(*) (i = 1; 2; 3; 4) coincide with 1=4 and the other two with 3=4. [35 marks]