Download presentation

Presentation is loading. Please wait.

Published byGianna Bartron Modified over 2 years ago

1
Auction Theory Class 5 – single-parameter implementation and risk aversion 1

2
2

3
Outline What objective function can be implemented in equilibrium? – Characterization result for single-parameter environments. Revenue effect of risk aversion. – Comparison of 1 st and 2 nd price auctions. 3

4
Implementation Many possible objective functions: – Maximizing efficiency; minimizing gaps in the society; maximizing revenue; fairness; etc. – Many exogenous constraints imply non-standard objectives. Problem: private information Which objectives can be implemented in equilibrium? – We saw that one can maximize efficiency in equilibrium. What about other objectives? We will show an exact characterization of implementable objectives. 4

5
Reminder: our setting Let v 1,…,v n be the private values (“types”) of the players (drawn from the interval [a,b]) Each player can eventually win or lose. – Winning gains the player a value of v i, losing gains her 0. – (More general than single-item auction.) An allocation function: Q:[a,b] n q 1,…,q n – q i = the probability that player i wins. Given an allocation function Q, let Q i (v i ) be the probability that player i wins. – In average, over all other values. 5

6
Characterization Recall that an auction consists of an allocation function Q and a payment function p. Theorem: An auction (Q,p) is truthful if and only if 1.(Monotonicity) Q i () is non-decreasing for every i. 2.(Unique payments) P i (v i )= v i ·Q i (v i ) – a vi Q i (x)dx Conclusion: only monotone objective functions are implementable. – Indeed, the efficient allocation is monotone (check!). 6 Theorem: An auction (Q,p) is truthful if and only if 1.(Monotonicity.) Q i () is non-decreasing for every i. 2.(Unique payments.) P i (v i )= v i ·Q i (v i ) – a vi Q i (x)dx

7
Reminder: our setting Proof: We actually already proved: truthfulness (monotonicity) + (unique payments) Let’s see where we proved monotonicity: 7

8
Proof Consider some auction protocol A, and a bidder i. Notations: in the auction A, – Q i (v) = the probability that bidder i wins when he bids v. – p i (v) = the expected payment of bidder i when he bids v. – u i (v) = the expected surplus (utility) of player i when he bids v and his true value is v. u i (v) = Q i (v) v - p i (v) In a truthful equilibrium: i gains higher surplus when bidding his true value v than some value v’. – Q i (v) v - p i (v) ≥ Q i (v’) v - p i (v’) 8 =u i (v’)+ ( v – v’) Q i (v’) =u i (v) We get: truthfulness u i (v) ≥ u i (v’)+ ( v – v’) Q i (v’)

9
Proof We get: truthfulness u i (v) ≥ u i (v’)+ ( v – v’) Q i (v’) or Similarly, since a bidder with true value v’ will not prefer bidding v and thus u i (v’) ≥ u i (v)+ ( v’ – v) Q i (v) or Let dv = v-v’ Taking dv 0 we get: 9 Given that v>v’

10
Rest of the proof We will now prove the other direction: if a mechanism satisfies (monotonicity)+(unique payments) then it is truthful. In other words: if the allocation is monotone, there is a payment scheme that defines a mechanism that implements this allocation function in equilibrium. Let’s see graphically what happens when a bidder with value v’ bids v>v’. 10

11
Proof: monotonicity truthfulness Proof: We saw that truthfulness is equivalent to: for every v,v’ : u i (v) - u i (v’) ≤ ( v – v’) Q i (v) We will show that (monotonicity)+(unique payments) implies the above inequality for all v,v’. (Assume w.l.o.g. that v>v’) We first show: Now, 11 Due to the unique-payment assumption Due to monotonicity

12
Single vs. multi parameter A comment: this characterization holds for general single-parameter domains – Not only for auction settings. Single parameter domains: a private value is one number. – Or alternatively, an ordered set. Multi-dimensional setting are less well understood. – Goal for extensive recent research. – We will discuss it soon. 12 Theorem: An auction (Q,p) is truthful if and only if 1.(Monotonicity.) Q i () is non-decreasing for every i. 2.(Unique payments.) P i (v i )= v i ·Q i (v i ) – a vi Q i (x)dx

13
Outline What objective function can be implemented in equilibrium? – Characterization result for single-parameter environments. Revenue effect of risk aversion. – Comparison of 1 st and 2 nd price auctions. 13

14
Risk Aversion We assumed so far that the bidders are risk-neutral. – Utility is separable (quasi linear), v i -p i Now: bidders are risk averse ( שונאי סיכון ). – All other assumptions still hold. We assume each bidder has a (von-Neumann- Morgenstern) utility function u(∙). – u is an increasing function (u’>0): u($10)>u(5$) – Risk aversion: u’’ < 0 14

15
Risk Aversion – reminder. 15 $5$10 u($5) u($10) ½*u($10) + ½*u($5) 7.5 u( ½* $10 + ½*$5 ) A risk averse bidder prefers the expected value over a lottery with the same expected value.

16
Auctions with Risk Averse Bidders The revenue equivalence theorem does not hold when bidders are risk averse. We would like to check: with risk-averse bidders, should a profit maximizing seller use 1 st -price or 2 nd -price auction? Observation: 2 nd -price auctions achieve the same revenue for risk-neutral and risk-averse bidders. – Bids are dominant-strategy, no uncertainty. 16

17
Auctions with Risk Averse Bidders Intuition: – risk-averse bidders hate losing. – Increasing the bid slightly increases their potential payment, but reduces uncertainty. Gain ε when you win, but risk losing v i -b i The equilibrium bid is higher than in the risk-neutral case. 17 Theorem: Assume that 1.Private values, distributed i.i.d. 2.All bidders have the same risk-averse utility u(∙) Then, E[1 st price revenue] ≥ E[2 nd price revenue]

18
1 st price + risk aversion: proof Let β(v) be a symmetric and monotone equilibrium strategy in a 1 st -price auction. Notation: let the probability that n-1 bidders have values of at most z be G(z), and G’(z)=g(z). – That is, G(z)=F(z) n-1 Bidder i has value v i and needs to decide what bid to make (denoted by β(z) ). – Will then win with probability G(z). Maximization problem: 18

19
1 st price + risk aversion: proof Proof: FOC: Or: But, since β(v) is best response of bidder 1, he must choose z=x: We didn’t use risk aversion yet… 19

20
1 st price + risk aversion: proof Fact: if u is concave, for all x we have (when u(0)=0) : 20 u(x) x xu’(x)

21
1 st price + risk aversion: proof For risk-averse bidders: With risk-neutral bidders (u(x)=x, u’(x)=1 for all x). Therefore, for every v 1 Since β(0) = b(0) =0, we have that for all v 1 21

22
1 st price + risk aversion: proof Summary of proof: in 1 st -price auctions, risk-averse bidders bid higher than risk-neutral bidders. Revenue with risk-averse bidders is greater. Another conclusion: with risk averse bidders, 22 Revenue in 1 st -price auctions Revenue in 2 nd -price auctions >

23
Summary We saw today: – Monotone objectives can be implemented (and only them) – Risk aversion makes sellers prefer 1 st -price auctions to 2 nd - price auctions. So far we discusses single-item auction in private value settings. – Next: common-value auctions, interdependent values, affiliated values. 23

Similar presentations

OK

Week 9 1 COS 444 Internet Auctions: Theory and Practice Spring 2008 Ken Steiglitz

Week 9 1 COS 444 Internet Auctions: Theory and Practice Spring 2008 Ken Steiglitz

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Topics for ppt on environmental science Ppt on mars one wiki Ppt on gir national park Convert pptx file into ppt online Ppt on indian army weapons pictures Ppt on object-oriented concepts and principles Ppt on airbag working principle of air Ppt on network security concepts Ppt on statistics and probability Ppt on nervous system