Strategy -1- Strategic equilibrium in auctions

Transcription

1 Strategy -- Strategic equilibrium in auctions A. Sealed high-bid auction 2 B. Sealed high-bid auction: a general approach 6 C. Other auctions: revenue equivalence theorem 27 D. Reserve price in the sealed high-bid auction 33 E. Dutch auction 35 F. Additional exercises 36 UCLA Auction House 38 pages

2 Strategy -2- A. Sealed high bid auction model Private information: Each buyer s value is private information. **

3 Strategy -3- Sealed high bid auction model Private information: Each buyer s value is private information. Common knowledge: It is common knowledge that buyer i s value, draw from a continuous distribution. We define F( v ) Pr{ v v }. i i i This is called the cumulative distribution function (c.d.f.). * v i, is an independent random

4 Strategy -4- Sealed high bid auction model Private information: Each buyer s value is private information. Common knowledge: It is common knowledge that buyer i s value is an independent random draw from a continuous distribution. We define F( v ) Pr{ v v }. i i i This is called the cumulative distribution function (c.d.f.). The values: The values lie on an interval [0, ].

5 Strategy -5- Strategies With private information a player s action depends upon his private information. In the sealed highbid auction, a player s private information is the value v i that he places on the item for sale. His bid is then some mapping b B ( v ) from every possible value i i i This mapping is the player s bidding strategy. Buyers with higher values have more to lose by not winning. So it is natural to assume that buyers with higher values will bid more so that Bv ( ) is a strictly increasing function. i i v i into a non-negative bid.

6 Strategy -6- Equilibrium strategies Since we assume that each buyer s value is a draw from the same distribution it is natural to assume that the equilibrium is symmetric. B ( v ) B( v ) i i i In a symmetric equilibrium, what is buyer s win probability if he has value ˆv? **

7 Strategy -7- Equilibrium strategies Since we assume that each buyer s value is a draw from the same distribution it is natural to assume that the equilibrium is symmetric. B ( v ) B( v ) i i i In a symmetric equilibrium, what is buyer s win probability if he has value ˆv? Key observation: Two buyers. Buyer wins if buyer 2 s bid is lower. Buyer 2 bids lower if and only if his value is lower. So buyer s equilibrium win probability is w( v ) Pr{ B( v2) B( v} Pr{ v2 v} F( v) *

8 Strategy -8- Equilibrium strategies Since we assume that each buyer s value is a draw from the same distribution it is natural to assume that the equilibrium is symmetric. B ( v ) B( v ) i i i Key observation: Two buyers. Buyer wins if buyer 2 s bid is lower. Buyer 2 bids lower only if his value is lower. So buyer s equilibrium win probability is w( v ) Pr{ B( v2) B( v} Pr{ v2 v} F( v) Three buyers. Buyer wins if and only if the other two buyers bids are both lower. This is true only if their values are both lower. So buyer s equilibrium win probability is w( v ) Pr{ v v } Pr{ v v } F( v ) 2 3 2

9 Strategy -9- Equilibrium Strategies Bayesian Nash Equilibrium (BNE) strategies: With private information, strategies that are mutual best response strategies are called Bayesian Nash Equilibrium strategies. Remark: With uncertainty about values, a strategy is a description of an action conditional upon a participants value. So conditional probabilities play a role. Symmetric BNE of the sealed high bid auction If all other buyers other then buyer i use the bidding strategy b B( v ) then buyer i s best j j response is to use the same strategy, i.e. b B( v ). i.e. equilibrium strategies are best responses. i i

10 Strategy -0- An example: Two buyers with values uniformly distributed on [0,00]. For the uniform distribution values are equally likely. Therefore 25 Pr{ vi 25}, Pr{ vi 50}, Pr{ vi 80}.. 00 vi Thus the c.d.f. is F( vi ) Pr( vi vi ) 00 For any guess as to the equilibrium strategy, We can check to see if the guess is correct. *

11 Strategy -- An example: Two buyers with values uniformly distributed on [0,00]. For the uniform distribution values are equally likely. Therefore 25 Pr{ vi 25}, Pr{ vi 50}, Pr{ vi 80}.. 00 vi Thus the c.d.f. is F( vi ) Pr( vi vi ) 00 For any guess as to the equilibrium strategy, We can check to see if the guess is correct. There are two buyers. Suppose that buyer 2 bids according to the strategy B( v ) v We need to show that buyer s best response is to bid b v. 2 Proposed equilibrium strategy Then these strategies are mutual best responses.

12 Strategy -2- Solving for buyer s best response when his value is v If buyer bids b ˆ 20 he has the high bid if B( v ) v bˆ 20, i.e. if v Buyer s win probability is therefore 40 wˆ ˆ. Pr{ v2 40} 00 Buyer s expected payoff is therefore 40 U( v ˆ ˆ, b) ( v b) w( b) ( v 20) 00

13 Strategy -3- If buyer bids b he has the high bid if B( v ) v b, i.e. v 2 2 b. Buyer s win probability is therefore 2b w( b) Pr{ v2 2 b} F(2 b). 00 Buyer s expected payoff is therefore 2b 2 U v b v b w b v b vb b (, ) ( ) ( ) ( ) ( ) b U 2 v b v b (, ) ( 2 ) 00 0 for a maximum. If buyer 2 uses the strategy b v we have therefore shown that buyer s expected gain is maximized if b v Thus the strategies are mutual best responses.

14 Strategy -4- Group Exercise: Three buyers with values uniformly distributed (a) Show that if buyer 2 and buyer 3 bid according to b j 2 v j, then buyer s best response is to bid b v 2 when his value is v (b) Show that for some 2, bj B( vj) vj is the equilibrium bidding strategy (c) What is the equilibrium bidding strategy with 4 buyers?

15 Strategy -5- Answer to (b) The probability that buyer wins with a bid of b is the joint probability that w ( b) Pr{ b b} Pr{ b b} 2 3 Pr{ v b} Pr{ v b} 2 3 b b b b Pr{ v2 } Pr{ v3 } F ( ) ( ) 2 2 b U( v, b) ( v b) w( b) ( v b)( ) ( vb 2 b ) b2 b and b3 b, ie. U b (2 2 vb 3 ) 0 2 b for a maximum. 2 Therefore buyer s best response is B ( v ) v. Note that this is true if Thus if the other buyers bid Bj( j) 3 j response is to do so as well. The problem with this approach is that it requires an inspired guess., then buyer s best

16 Strategy -6- B. Sealed high bid auctions: A general approach - - Characterize equilibrium payoffs U( v) w( v)( v B( v)) equilibrium equilibrium net gain payoff win probability if buyer wins Rewrite as follows U( v) w( v) v w( v) B( v) equilibrium expected expected payoff gross gain buyer payment *

17 Strategy -7- B. Sealed high bid auctions: A general approach - - Characterize equilibrium payoffs U( v) w( v)( v B( v)) equilibrium equilibrium net gain payoff win probability if buyer wins Rewrite as follows U( v) w( v) v w( v) B( v) (*) equilibrium expected expected payoff gross gain buyer payment We solve (i) for the equilibrium win probability and hence the expected gross gain, (ii) for the equilibrium payoff. We can then solve for the equilibrium bid function by appealing to (*).

18 Strategy -8- Remark: We can write the buyer s equilibrium payoff as U( v) w( v) v w( v) B( v) equilibrium expected expected payoff gross gain buyer payment Since the payment by the buyer is the revenue that the seller receives from the buyer, U( v) w( v) v R( v) equilibrium expected expected payoff gross gain seller revenue (This will be useful when we consider other auctions.)

19 Strategy -9- Solution: Step : Obtain an expression for buyer s equilibrium win probability Class Exercise: Two buyers: Why is the equilibrium probability w( v) F( v)? Three buyers: Why is the equilibrium probability w( v ) ( ) 2 F v?

20 Strategy -20- Equilibrium win probability with I buyers If buyer with a value of v makes an equilibrium bid of Bv ( ), then she has the high bid if b B( v ) B( v ), for j 2,..., I j j equivalently, vj v, for j 2,..., I The joint probability of this event is w( v ) Pr{ v v } Pr{ v v }... Pr{ v v } F( v ) I 2 3 I

21 Strategy -2- Step 2: Suppose that buyer is naïve (stupid!) Suppose buyer uses the strategy BN( v ) B( vˆ ) b ˆ regardless of her value Since Bv ( ) is her best response strategy, The naïve strategy is a best response If and only if v vˆ. (Note that, regardless of her value, buyer bids ˆb. Then her win probability is wv ( ˆ ).)

22 Strategy -22- Let Uv ( ) be her equilibrium payoff. Let UN ( v) be her payoff if she is naive. Then U ( vˆ) U ( vˆ) N and U( v ) UN( v ) for all v vˆ Key conclusion: The graphs of the two functions must be tangential* at v vˆ U ( vˆ) U ( vˆ) N * As we shall next show, the graph of UN ( v ) is actually a line.

23 Strategy -23- The naïve buyer always bids ˆb so her win probability is always Therefore wv ( ˆ ) UN( v ) w( vˆ )( v bˆ) w( vˆ ) v w( vˆ ) bˆ. *

24 Strategy -24- But the naïve buyer always bids ˆb so her win probability is always Therefore wv ( ˆ ) UN( v ) w( vˆ )( v bˆ) w( vˆ ) v w( vˆ ) bˆ This is a line of slope wv ( ˆ ). Therefore U ( vˆ ) U ( vˆ ) w( vˆ ). N w( vˆ) v R( vˆ). Since we can make the same argument for any ˆv, we have proved the following result. Proposition: Marginal equilibrium payoff The rate at which the equilibrium payoff rises with the buyer s value is equal to the equilibrium win probability U ( v) w( v)

25 Strategy -25- The lowest value participant has an equilibrium payoff of zero since other buyers have higher values with probability. Thus they make higher bids with probability. We can then integrate U () v to obtain the equilibrium payoff Uv (). v i U( v ) w( v) dv where i 0 w( v) F( v) I Also U( v ) w( v )( v B( v )) i i i i equilibrium equilibrium net gain payoff win probability if buyer wins Then we can solve for the equilibrium bid function.

26 Strategy -26- Example: Uniform distribution with 3 buyers Proposition: U ( v) w( v) F( v) where U( v) w( v)( v B( v)) Suppose that buyers values are uniformly distributed on [0,] (in millions of dollars) so F() v 2 2. U ( v) F( v) v Then 3 U() v 3 v k But the lowest type wins with zero probability so U(0) 0. Therefore U() v Also 3 v 3 2 U( v) w( v)( v B( v)) v ( v B( v)). v Therefore 2 3 v ( v B( v)) 3 v and so v B() v 3 v. 2 Therefore B() v 3 v.

27 Strategy -27- C. Other sealed bid auctions: We will now generalize the above argument. In the sealed high bid auction U( v) w( v)( v B( v)) equilibrium equilibrium net gain payoff win probability if buyer wins This can be rewritten as follows U( v) w( v) v w( v) B( v) equilibrium expected expected payoff gross gain buyer payment Since the payment by the buyer is the revenue of the seller we can also write U( v) w( v) v R( v) equilibrium expected expected payoff gross gain seller revenue

28 Strategy -28- Consider some other auction in which the equilibrium bidding strategies are strictly increasing Example : Sealed second-bid auction Example 2: All pay auction Buyers submit non-refundable bids. The high bidder is the winner. Then the equilibrium win probability is w( v ) F( v ) I i i, just as in the sealed high-bid auction. Let Rv ( i ) be the equilibrium expected payment by buyer i. Then buyer i s equilibrium payoff is U( v ) w( v ) v R( v ) i i i i equilibrium expected expected payoff gross gain seller revenue We can argue step 2 in the same way.

29 Strategy -29- Equilibrium payoff: U( v ) w( v ) v R( v ) i i i i Step 2: Suppose that buyer is naïve (stupid!) B N( v ) ( ˆ ) Suppose buyer uses the strategy B v regardless of her value. Then her expected payment is Rv ( ˆ ), regardless of her value U v w vˆ v R vˆ N( ) ( ) ( ) Since Bv ( ) is her best response strategy, the naïve strategy is a best response If and only if v vˆ. U v w vˆ v R vˆ N( ) ( ) ( ) Is a line with slope wv ( ˆ ). Therefore U ( vˆ ) U N ( vˆ ) w( vˆ ).

30 Strategy -30- Since we can make the same argument for any ˆv, we have proved the following result. Proposition: The incremental equilibrium payoff for a buyer with a higher value is equal to the equilibrium win probability U ( ) w( ) **

31 Strategy -3- Since we can make the same argument for any ˆv, we have proved the following result. Proposition: The incremental equilibrium payoff for a buyer with a higher value is equal to the equilibrium win probability U ( ) w( ) A buyer with the lowest value has a zero probability of winning so U(0) 0. Therefore Proposition: Buyer equivalence Theorem The equilibrium payoff for a buyer with value v i is v i U( v ) w( v) dv i 0 *

32 Strategy -32- Since we can make the same argument for any ˆv, we have proved the following result. Proposition: The incremental equilibrium payoff for a buyer with a higher value is equal to the equilibrium win probability U ( ) w( ) A buyer with the lowest value has a zero probability of winning so U(0) 0. Therefore Proposition: Buyer equivalence Theorem The equilibrium payoff for a buyer with value v i is v i U( v ) w( v) dv i 0 Since U( v ) w( v ) v R( v ) i i i i Proposition: Payment equivalence Theorem The equilibrium expected payment by a buyer, Rv ( i ) is the same as in the sealed high-bid auction. Revenue equivalence: Since the expected payment is the expected revenue of the seller, the expected seller revenue is the same.

33 Strategy -33- D. Reserve price in the sealed high-bid auction Proposition: U ( v) w( v) where U( v) w( v)( v B( v)) Analysis: The seller sets a reserve price (i.e. minimum bid of than v is Fv (). What is the equilibrium payoff? v 0). The probability that a buyer s value is less A buyer has a value vi v0 has a strictly positive payoff if he enters the auction and make a bid satisfying v0 B( v ) v. * i i

34 Strategy -34- D. Reserve price in the sealed high-bid auction Proposition: U ( v) w( v) where U( v) w( v)( v B( v)) Analysis: The seller sets a reserve price (i.e. minimum bid of than v is Fv (). What is the equilibrium payoff? v 0). The probability that a buyer s value is less A buyer with a value vi v0 enters the auction and make a bid satisfying v0 B( v ) v. i i This is depicted opposite. It follows that v0 B( v0) 0. Hence U( v0 ) w( v0 )( v0 B( v0 )) 0 Therefore U( v ) U( v ) U( v ) U ( v) dv w( v) dv i i 0 v i i v0 v0 v

35 Strategy -35- E. Open descending price auction (descending clock auction) Clock starts ticking down. When a buyer raises his hand the clock stops and the buyer pays the price on the clock. Proposition: The equilibrium bidding strategy is to stop the clock when the asking price is the equilibrium bid in the sealed high-bid auction

36 Strategy -36- E. Exercises The first two exercises consider the following model discussed on page 8. If the total output is q q2... qn then the market clearing price is p( q... q n ). Suppose n n i i, Ci ( qi ) 2q i i i p( q ) 60 q, i=,,n Exercise : Nash equilibrium with more than two firms What is the equilibrium price if there are (i) three identical firms (ii) 5 identical firms Exercise 2: Nash equilibrium with a large number of firms Show that the equilibrium price approaches 2 as the number of firms grows large.

37 Strategy -37- Exercise 3: Equilibrium with increasing marginal cost Suppose p( q) 60 ( q q2), C ( q ) 50 q and 2 C ( q ) 50 q (a) Solve for the Nash Equilibrium outputs and price. (b) What would the firms do if they could collude and so maximize the total profit of the two firms? (c) Might the answer change if C ( q ) 00 q

38 Strategy -38- Exercise 5: Pricing game If a firm cannot change its capacity quickly then the quantity setting model makes sense. But what if capacity can be easily changed. Then a firm can lower a price and still guarantee delivery, or raise a price and sell off unused capacity. C ( q ) c q where c 4 and c2 7. Demands are f f f f q p 0 p, q p 20 p2. 2 Then the profit of firm f is ( p, q ( p)) p q ( p) C ( q ( p)) ( p c ) q ( p) f f f f f f f f f f (a) For each firm solve for the best response function p b ( q2 ) and p2 b2 ( p ). (b) Depict these in a neat figure. What are the equilibrium prices in this game? (c) If firm 2 produces nothing what will firm do? Plot a price vector p ( p, p ) indicating the monopoly outcome with only firm producing. Hint: The price of commodity 2 must be chosen so that demand for firm 2 s output is zero. (d) Starting from this price pair, examine the adjustment process proposed by Cournot. (e) Compare the equilibrium profits with those in the quantity setting game.

39 Strategy Sealed high-bid auction with a reserve price (minimum price). Each of two buyers has a value that is uniformly distributed on [0,]. The seller sets a reserve price of. (a) Explain why U () v v for all v. (b) Explain why Uv ( ) 0 for all v. (c) Hence solve for Uv (). (d) Use this result to solve for the equilibrium bid function.