payoff) while M gives 1 irrespective of player 1’s strategy.
Therefore, M is eliminated by mixing L and R .
After eliminating M , we can further eliminate D (step 2) and L
(step 3), eventually picks up ( U , R ) as a unique outcome.
M P 1 : max
q≥0 p(q)q − c(q).
where p(q)q is a revenue and c(q) is a cost when the output is fixed to q. Let π(q) = p(q)q − c(q) denote the revenue function. Assume that the firm’s objective function π(q) is convex and differentiable. Then, the first order condition is:
1. continuous at a point x 0 if, for all ε > 0, there exists δ > 0
such that d(x, x 0 ) < δ implies that d(f (x), f (x 0 )) < ε.
2. continuous if it is continuous at every point in its domain.
3. uniformly continuous if, for all ε > 0, there exists δ > 0 such
Klemperer (2002), “How (not) to Run Auctions: The European 3G Telecom Auctions,” European Economic Review. Milgrom (2004) Putting Auction Theory to Work Cambridge U Press[r]
Randomized Strategies
No strategy looks to be dominated…
If a player 2 randomizes L and R with 50% each, then
Such mixed (randomized) strategy yields 1.5 (as an expected payoff) while M gives 1 irrespective of player 1’s strategy.
3. Auction (14 points)
Suppose that a seller auctions one object to two buyers, = 1, 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays the average of the two bids b + b
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.
(b) If consumer’s choice satis…es the weak axiom of revealed preferences, we can always construct a utility function which is consistent with such choice behav- iour.
(c) If a consumer problem has a solution, then it must be unique whenever the consumer’s preference relation is convex.
(a) Characterize the first-best solution.
(b) Suppose that the seller cannot observe θ: θ ∈ {θ L , θ H } and Pr[θ = θ L ] = β with
0 < θ L < θ H . Set up the seller’s optimization problem under this asymmetric
information structure.
Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’s short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’s
If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]
3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]