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

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

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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

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

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

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

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

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’)

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’

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

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

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

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

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

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.

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

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]

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

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

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)

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

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 >

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

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