Theory of Auctions. Carlos Hurtado. Jun 23th, Department of Economics University of Illinois at Urbana-Champaign

Transcription

1 Theory of Auctions Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign Jun 23th, 2015 C. Hurtado (UIUC - Economics) Game Theory

2 On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

3 Formalizing the Game On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 1 / 28

4 Formalizing the Game Formalizing the Game A game with incomplete information G = (Θ, S, P, u) consists of: 1. A set Θ = Θ 1x xθ N, where Θ i is the (finite) set of possible types for player i. 2. A set S = S 1x xs N, where S i is the set of possible strategies for player i. 3. A joint probability distribution p(θ 1,, Θ N ) over types. For finite type space, assume that p(θ i) > 0 for all θ i Θ i. 4. Payoff functions u i : SxΘ R. Remark: Note that payoffs can depend not only on your own type, but on your rivals types. If u i depends on θ i, but not on θ i, we sometimes say the game has private values. In order to analyze these types of games, we rely on a fundamental (and Nobel-prize winning) observation by Harsanyi (1968): Games of incomplete information can be thought of as games of complete but imperfect information where nature makes the first move (selecting θ 1,, θ N ), but not everyone observes nature s move (i.e. player i learns θ i but not θ i). C. Hurtado (UIUC - Economics) Game Theory 2 / 28

5 Formalizing the Game Formalizing the Game A Bayesian pure strategy for player i in a Bayesian game is a function s i(θ i), or decision rule, that gives the player s strategy choice for each realization of his type θ i. That is, s i : Θ i S i. We write S Θ i for the set of Bayesian pure strategies. Player i s expected payoff given a profile of Bayesian pure strategies for players (s 1(θ 1),, s N (θ N )) is given by: Definition ũ i(s i(θ i), s i(θ i)) = E Θ [u i(s i(θ i), s i(θ i), θ i, θ i)] A Pure Strategy Bayesian Nash Equilibrium (PSBNE) of the game G = (Θ, S, P, u) is a profile of decision rules (s 1(θ 1),, s N (θ N )) such that for all s i(θ i) S Θ i. ũ i(s i(θ i), s i(θ i)) ũ i( s i(θ i), s i(θ i)) An alternative definition for PSBNE is: s i(θ i) arg max u i(s i(θ i), s i(θ i), θ i, θ i) p(θ i, θ i) s i S Θ i θ Θ C. Hurtado (UIUC - Economics) Game Theory 3 / 28

6 On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 4 / 28

7 Consider the following two-player game of incomplete information: 1/2 L R T v,1 0,2 B,3 1,1 1 2 Player 1 s value v is his own private information. It is common knowledge that v is a random variable that is uniformly distributed on [0, 1]. Determine all pure strategy Bayesian-Nash equilibria. Be sure to clarify how you know that you have found all of the equilibria. Hint: Organize your answer according to the possible strategies of player 2. Two Hints: (i) player 1 must base his strategic choice on his private information v; (ii) if it is common knowledge that v is uniformly distributed on [0, 1], then you do not have the freedom to specify player 2 s beliefs about the value of v as part of defining the equilibrium. C. Hurtado (UIUC - Economics) Game Theory 5 / 28

8 1/2 L R T v,1 0,2 B,3 1,1 1 2 We first try to construct an equilibrium in which 2 plays L. In this case, player 1 s best response is to choose B if v 1/2 and T if v 1/2; with this strategy for 1, player 2 receives an expected payoff of 2 from L and 3/2 from R, and so L is his best response. The equilibrium is therefore: 2 : L { B 1 : T if v < 1/2 if v 1/2 We next try to construct an equilibrium in which 2 plays R. In this case, player 1 s best response is to choose B for all values of v. If 1 chooses B, however, then 2 s best response is L. Consequently, there is no Bayesian Nash equilibrium in which 2 chooses R. C. Hurtado (UIUC - Economics) Game Theory 6 / 28

9 Consider the following two-player game of incomplete information: 1/2 L R 1 T v,0,2 2 1 B, 3 v,1 2 2 Player 1 s value v is his own private information. It is common knowledge that v is a random variable that is uniformly distributed on [0, 1]. Notice that v appears in both the T,L outcome and the B,R outcome. Determine all pure strategy Bayesian-Nash equilibria. Be sure to clarify how you know that you have found all of the equilibria. Hint: Organize your answer according to the possible strategies of player 2. C. Hurtado (UIUC - Economics) Game Theory 7 / 28

10 1/2 L R 1 T v,0,2 2 1 B, 3 v,1 2 2 We first try to construct an equilibrium in which 2 plays L. In this case, player 1 s best response is to choose B if v 1/2 and T if v 1/2; with this strategy for 1, player 2 receives an expected payoff of 3/4 from L and 3/2 from R, and so R is his best response. We therefore cannot construct a pure strategy equilibrium in which 2 plays L. We next try to construct an equilibrium in which 2 plays R. In this case, player 1 s best response is to choose B if v 1/3 and T if v 1/3; with this strategy for 1, player 2 s expected payoff is 1 from playing L and 4/3 from playing R. He therefore plays R. The equilibrium is therefore 2 : R { T 1 : B if v < 1/3 if v 1/3 C. Hurtado (UIUC - Economics) Game Theory 8 / 28

11 Consider the following two-player game of incomplete information: 1/2 C NC C 1 θ 1,1 θ 2 1 θ 2,1 NC 1,1 θ 1 0,0 Player i s value θ i is his own private information. It is common knowledge that the θ i are independent random variables uniformly distributed on [0, 2]. Determine all pure strategy Bayesian-Nash equilibria. Be sure to clarify how you know that you have found all of the equilibria. C. Hurtado (UIUC - Economics) Game Theory 9 / 28

12 1/2 C NC C 1 θ 1,1 θ 2 1 θ 2,1 NC 1,1 θ 1 0,0 Note that, if player i s value is θ i and C is the best strategy, then C is the best strategy for any θ i < θ i. This suggest that the strategy for player i is { C θ i θi s i(θ i) = NC θ i > θi Then, for player i we have E Θ [C] = (1 θ i) p(θ i θ i) + (1 θ i)p(θ i > θ i) and E Θ [NC] = p(θ i θ i). In equilibrium each player should be indifferent between his two choices at this value of his type: ( (1 θ i) θ i 2 + (1 θ i) 1 θ i 2 ( (1 θ i) 1 θ i 2 ) ) = θ i 2 = θ i θ i 2 C. Hurtado (UIUC - Economics) Game Theory 10 / 28

13 1/2 C NC C 1 θ 1,1 θ 2 1 θ 2,1 NC 1,1 θ 1 0,0 Also notice that, in equilibrium, the agent i must be indifferent when θ i = θi, then, ( ) ( ) (1 θ 2 ) 1 θ 2 2 = (1 θ 1 ) 1 θ 1 2 which implies that Hence, Which implies θ 1 = θ 2 θ 2 i = 2 3θ i + θ 2 i θ i = 2 3 C. Hurtado (UIUC - Economics) Game Theory 11 / 28

14 1/2 C NC C 1 θ 1,1 θ 2 1 θ 2,1 NC 1,1 θ 1 0,0 Then, the unique Pure Strategies Bayesian Nash Equilibrium (PSBNE) is: s i(θ i) = { C NC θ i 2 3 θ i > 2 3 C. Hurtado (UIUC - Economics) Game Theory 12 / 28

15 Two firms jointly share their research outputs. They might be envisioned as divisions of the same firm. Each firm can independently choose to spend c (0, 1) to develop the zigger, a device that is then made available to the other firm. Firm i s type is θ i, which is believed by firm i to be independently drawn from the uniform distribution on [0, 1]. The benefit of the zigger when the type is θ i is θ 2 i. The timing is: the two firms privately observe their own type. Then they each simultaneously choose either to develop the zigger or not. Solve for the Pure Strategy Nash Equilibrium. C. Hurtado (UIUC - Economics) Game Theory 13 / 28

16 value of the zigger to firm i: θ 2 i payoff if the zigger is not provided: 0 payoff if it builds the zigger: θ 2 i c payoff if it does not build the zigger but firm i does: θ 2 i s i : [0, 1] {yes, no} Let p i denote the probability that firm i produces the zigger, given its strategy s i. Firm i should provide the zigger only if Equivalently, θ 2 i c p i θ 2 i θ i Hence, firm i and i use a cutoff strategy. c 1 p i C. Hurtado (UIUC - Economics) Game Theory 14 / 28

17 Let θ i be the cutoff point. Then, firm i will provide the zigger with probability p i = 1 θi c c = 1 = 1 1 p i θ i Therefore That is, and symmetrically, Canceling, θi c = θ 2 i That is, the only PSBNE is symmetric. θ i θ i = c θ 2 i θ i θ i = c = θ i C. Hurtado (UIUC - Economics) Game Theory 15 / 28

18 Substituting into an equation above implies θ i = θ i = c 1 3 Where is the cost of free riding? The zigger should be provided by one of the two firms if θ 2 i c Given that c (0, 1), we have that c 1 2 < c 1 3 C. Hurtado (UIUC - Economics) Game Theory 16 / 28

19 Sealed Bid (First-Price) Auction On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 17 / 28

20 Sealed Bid (First-Price) Auction Sealed Bid (First-Price) Auction In a sealed bid, or first price, auction, bidders submit sealed bids b 1,..., b N. The bidders who submits the highest bid is awarded the object, and pays his bid. Under these rules, it should be clear that bidders will not want to bid their true values. By doing so, they would ensure a zero profit. By bidding somewhat below their values, they can potentially make a profit some of the time. Model: - Bidders i = 1,..., N - Bidder i observes a signal θ i F ( ) - Bidders signals are in [θ, θ] for all bidders. - Bidders signals are independent - Bidder i s value of the object is v i(θ i) = θ i - Bidders bids depend on the signal θ i with the same functional form b( ) If Bidder i wins the auction he has an utility of v i(θ i) b(θ i) The probability of wining the auction is Pr[b i = b(θ i) b i = b(θ i)] = Pr[θ i b 1 (b i)] C. Hurtado (UIUC - Economics) Game Theory 18 / 28

21 Sealed Bid (First-Price) Auction Sealed Bid (First-Price) Auction To ensure that b 1 ( ) exist, we impose that each bidder uses a bid strategy that is a strictly increasing and continuous. Then, the bidder i s expected payoff, as a function of his bid b i and signal θ i is: that can be reduced to (why?) U(b i, θ i) = (v i(θ i) b(θ i)) Pr[θ i b 1 (b i)] U(b i, θ i) = (θ i b i) F n 1 [b 1 (b i)] Bidders would like to optimize their bid to maximize their expected utility: U(b i, θ i ) = (θ i b i ) (n 1) F n 2 [b 1 (b i )] f (b 1 1 (b i )) b i b (b 1 (b i )) F n 1 [b 1 (b i )] = 0 Here we need to assume further that b( ) is differentiable. C. Hurtado (UIUC - Economics) Game Theory 19 / 28

22 Sealed Bid (First-Price) Auction Sealed Bid (First-Price) Auction This can be reduced to the following differential equation (why?): b (θ i ) = (θ i b(θ i )) (n 1) f (θ i ) F [θ i ] The solution to this differential equation is the symmetric equilibrium. Another aproach: U(b(θ i), θ i) = U(θ i) = (θ i b(θ i)) Pr[θ i θ i] Because i is playing a best-response in equilibrium: Applying the envelope theorem U(θ i) = max(θ i b i) F n 1 (b 1 (b i)) b i du(θ i) dθ i = F n 1 (b 1 (b(θ i))) = F n 1 (θ i) C. Hurtado (UIUC - Economics) Game Theory 20 / 28

23 Sealed Bid (First-Price) Auction Sealed Bid (First-Price) Auction Then, imply U(θ i) = du(θ i) dθ i = F n 1 (θ i) θ i F n 1 (θ)dθ + K Given that U(θ) = 0 we have K = 0. Then, U(θ i) = (θ i b(θ i)) F n 1 (θ i) = θ θ i F n 1 (θ)dθ Hence: b(θ i) = θ i θi θ θ F n 1 (θ)dθ F n 1 (θ i) C. Hurtado (UIUC - Economics) Game Theory 21 / 28

24 Vickrey (Second-Price) Auction On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 22 / 28

25 Vickrey (Second-Price) Auction Vickrey (Second-Price) Auction Theorem In a Vickrey, or second price, auction, bidders are asked to submit sealed bids b 1,..., b N. The bidder who submits the highest bid is awarded the object, and pays the amount of the second highest bid. In a second price auction, it is a weakly dominant strategy to bid one s value, b i(θ i) = θ i Proof: Suppose i s value is θ i, and she considers bidding b i > θ i. Let ˆb denote the highest bid of the other bidders j i (from i s perspective this is a random variable). There are three possible outcomes from i s perspective: 1) ˆb > b i > θ i 2) b i > ˆb > θ i 3) b i > θ i > ˆb C. Hurtado (UIUC - Economics) Game Theory 23 / 28

26 Vickrey (Second-Price) Auction Vickrey (Second-Price) Auction Proof: case 1 or 3: Player i would have done equally well to bid θ i rather than b i > θ i. (why?) - In case 1: ˆb > b i > θ i, she won t win regardless. - In case 3: b i > θ i > ˆb, she win independently by bidding b i or θ i case 2: b i > ˆb > θ i - she will win, and will pay ˆb regardless. - Player i will win and pay more than her value if she bids ˆb, something that won t happen if she bids θ i Thus, i does better to bid θ i than b i > θ i. A similar argument shows that i also does better to bid θ i than to bid b i < θ i C. Hurtado (UIUC - Economics) Game Theory 24 / 28

27 Exercises On the Agenda 1 Formalizing the Game 2 3 Sealed Bid (First-Price) Auction 4 Vickrey (Second-Price) Auction 5 Exercises C. Hurtado (UIUC - Economics) Game Theory 25 / 28

28 Exercises Exercises Consider the following two-player game of incomplete information: 1/2 T B T 2θ 1,3θ 2 1,1 C 1,0 0,1 Player i s value θ i is his own private information. It is common knowledge that the θ i are independent random variables uniformly distributed on [0, 1]. Determine all pure strategy Bayesian-Nash equilibria. Be sure to clarify how you know that you have found all of the equilibria. C. Hurtado (UIUC - Economics) Game Theory 26 / 28

29 Exercises Exercises Consider an economy in which there are two consumers, a public good x and a private good y. Consider the voluntary contribution game in which each player i contributes an amount z i of his private good whereupon an amount x = z 1 + z 2 of the public good is produced. Consumer i has large initial endowment ŷ i of y and a utility function u i(x, y i) = ax + x yi where y i = ŷ i z i. Consumer i s preference parameter a i is either 4 or 6, but only consumer i knows which. The other consumer believes that a i = 4 with probability 1/2 and a i = 6 with probability 1/2. Compute a Bayesian Nash equilibrium of this game in which each player uses the same strategy (i.e., a symmetric Bayesian Nash equilibrium). C. Hurtado (UIUC - Economics) Game Theory 27 / 28

30 Exercises Exercises The market of lemons: Consider a seller of a used car and a potential buyer of that car. Suppose that quality of the car, θ, is a uniform draw from [0, 1]. This quality is known to the seller, but not to the buyer. Suppose that the buyer can make an offer p [0, 1] to the seller, and the seller can then decide whether to accept or reject the buyer s offer. Payoffs are as follows: u S = { p θ if offer accepted if offer rejected u B = { a + bθ p 0 if offer accepted if offer rejected Assume that a [0, 1), that b (0, 2), and that a + b > 1. These assumptions imply that for all θ, it is more efficient for the buyer to own the car. Find the unique PSBNE of this game. C. Hurtado (UIUC - Economics) Game Theory 28 / 28