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

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

**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.

**Reminder: our setting** • Let v1,…,vn 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 vi, losing gains her 0. • (More general than single-item auction.) • An allocation function: Q:[a,b]nq1,…,qn • qi= the probability that player i wins. • Given an allocation function Q, let Qi(vi) be the probability that player i wins. • In average, over all other values.

**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 • (Monotonicity) Qi() is non-decreasing for every i. • (Unique payments) Pi(vi)=vi·Qi(vi) – aviQi(x)dx Conclusion: only monotone objective functions are implementable. • Indeed, the efficient allocation is monotone (check!). Theorem: An auction (Q,p) is truthful if and only if • (Monotonicity.) Qi() is non-decreasing for every i. • (Unique payments.) Pi(vi)=vi·Qi(vi) – aviQi(x)dx

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

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

**Proof** • We get: truthfulness ui(v)≥ ui(v’)+ ( v – v’) Qi(v’) or • Similarly, since a bidder with true value v’ will not prefer bidding v and thus ui(v’)≥ ui(v)+ ( v’ – v) Qi(v) or Let dv = v-v’ Taking dv 0 we get: 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’.

**Proof: monotonicity truthfulness** Proof: We saw that truthfulness is equivalent to:for every v,v’ : ui(v)- ui(v’) ≤ ( v – v’) Qi(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, Due to the unique-payment assumption Due to monotonicity

**Single vs. multi parameter** Theorem: An auction (Q,p) is truthful if and only if • (Monotonicity.) Qi() is non-decreasing for every i. • (Unique payments.) Pi(vi)=vi·Qi(vi) – aviQi(x)dx 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.

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

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

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

**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 1st-price or 2nd-price auction? • Observation: 2nd-price auctions achieve the same revenue for risk-neutral and risk-averse bidders. • Bids are dominant-strategy, no uncertainty.

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

**1st price + risk aversion: proof ** • Let β(v) be a symmetric and monotone equilibrium strategy in a 1st-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 viand needs to decide what bid to make (denoted by β(z) ). • Will then win with probability G(z). • Maximization problem:

**1st 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…

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

**1st price + risk aversion: proof ** For risk-averse bidders: With risk-neutral bidders (u(x)=x, u’(x)=1 for all x). Therefore, for every v1 Since β(0) =b(0) =0, we have that for all v1

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

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