Players can reach Nash equilibrium only by rational reasoning in some games, e.g., Prisoners’ dilemma. However, rationality alone is often insufficient to lead to NE. (see Battle of the sexes, Chicken game, etc.) A correct belief about players’ future strategies combined with rationality is enough to achieve NE. 2 - 4 help players to share a correct belief.
(b) We will get (B; Z) in the following iterated elimination process: Step 1: We can erase X since X is strictly dominated by Z.
Step 2: Given step 1, we can erase A since A is strictly dominated by B.
Step 3: Given steps 1 and 2, we can erase Y since Y is strictly dominated by Z. (c) Any combinations of x and y that satisfy x + y = 100 are Nash equilibria. Clearly, there are 101 such equilibria, i.e., (0; 100)(1; 99):::(100; 0).
j + x j − x i x j , where x i is i’s effort and x j is the effort of the other player. Assume
x 1 , x 2 ≥ 0.
(a) Find the Nash equilibrium of this game. Is it Pareto efficient?
(b) Suppose that the players interact over time, which we model with the infinitely repeated version of the game. Let δ denote the (common) discount factor of the players. Under what conditions can the players sustain some positive effort level k = x 1 = x 2 > 0 over time?
Proof of Pratt’s Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by
(nw1) means student s prefers an empty slot at school c to her own assignment, and (nw2) and (nw3) mean that legal constraints are not violated when s is assigned the empty slot without changing other students’ assignments.
The second property is about no-envy, which is also widely used in the context of school choice. But due to the structure of controlled school choice, as in Definition 1, even when a student prefers a school to her own and there is a student with lower priority in the school, the envy is not justified if the student’s move violates the legal constraints. Definition 2 formally states the condition for a student to have justified envy.
Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’s payo¤ is the same in every Nash equilibrium.
4. Auctions (30 points)
Suppose that the government auctions one block of radio spectrum to two risk neu- tral mobile phone companies, i = 1, 2. The companies submit bids simultaneously, and the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:
(d) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(e) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c 1 = c 2 = 8. Then, derive the
Hint: Your answers in (a) – (c) may change depending on the value of θ.
4. Duopoly (20 points)
Consider a duopoly game in which two firms, denoted by firm 1 and firm 2, simul- taneously and independently select their own price, p 1 and p 2 . The firms’ products are differentiated. After the prices are set, consumers demand 24 − p i +
Three firms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.
However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’s axioms is in doubt.
where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X.
This assumes the case of private values in which player i’s payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by
Let w = (w 1 , w 2 , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors 1 and 2. (c) Suppose w 3 >