Review! November 28, 2011 () November 28, 2011 1 / 23 Mechanism - - PowerPoint PPT Presentation

Review!

November 28, 2011

() November 28, 2011 1 / 23

Mechanism Design

Design mechanisms/auctions such that when participants play selfishly, the designer’s goals are achieved. Some typical settings: markets: Given individual preferences for goods and money, determine the right way to reallocate the goods and money, e.g to maximize social welfare auctions: given buyers with preferences over items being sold, determine winners of auction and payments so as to maximize auctioneer’s profit. resource allocation in distribution systems: given resource owners that incur costs when their resources are used, and users with different needs and willingness to pay for resources, determine the allocation of resources to user to optimize global objective function.

() November 28, 2011 2 / 23

Single Parameter Allocation Problems

Auctioneer offering/allocating good/service. n agents, agent i has value vi for receiving good. There are constraints on which subsets of agents can simultaneously be

• served. For example:

single item auctions, k-unit auctions digital goods auctions ad auctions multiple markets – can only sell in one of them

The auction takes as input a bid bector b1, . . . , bn and chooses as output a feasible subset S of winning bidders (specified by allocation vector x) and a price pi (≤ bi) for each i ∈ S. Agents bid to maximize their own utility ui = vixi − pi. Goal: Design auction to achieve goals, which are usually: profit maximization, i.e. maximize

i pi

social welfare maximization, i.e. maximize

i vixi.

() November 28, 2011 3 / 23

Strategies and Equilibria

Definition A strategy for an agent in an auction is a mapping from values to bids (or actions), i.e. si(vi) describes how player i plays when his value is vi. Definition Dominant Strategies: An auction has dominant strategies (s1, . . . sn) if for all i and all b−i, agent i’s utility is maximized by playing si(vi). When the dominant strategy si(vi) = vi, we say the auction is truthful. Theorem The Revelation Principle: For any auction with dominant strategies, there is an equivalent direct-revelation auction which results in same outcome and payments and is truthful.

() November 28, 2011 4 / 23

Characterization of Truthful Auctions

Definition An auction is truthful if it is a dominant strategy for each player to bid their true value. Also, Let xi(v) be the probability that agent i wins when players values are v. let pi(v) be expected payment of agent i when players values are v. (Monotonicity:) An auction is truthful iff for any fixed v−i, xi(vi, v−i) is monotone non-decreasing in vi. For deterministic mechanisms, this means that there is a threshold t(v−i) such that xi(vi, v−i) is 1 when vi > t(v−i) and xi(vi, v−i) is 0 when vi < t(v−i). (Allocation rule determines payments:) pi(v) = vixi(v) − ∞ xi(w, v−i)dw + pi(0, v−i). For deterministic mechanisms, the payment when the player wins is the threshold bid t(v−i).

() November 28, 2011 5 / 23

Truthful auction examples

Vickrey (Second Price) Auction for a single item: Sell to bidder with the highest bid (= by assumption his value); Payment: 2nd highest bid. k-item auction: sell to bidders with top k bids; Payment: k+1st bid. Social welfare maximization: Choose feasible subset S of bidders such that

i∈S vi maximized. S are the winners.

Payment of bidder j ∈ S is:

i∈S′ vi − i∈S\j vi where S′ is the feasible

subset of bidders excluding j such that

i∈S′ vi maximized.

() November 28, 2011 6 / 23

Bayes-Nash Equilibrium

Assume players values are drawn from known prior distributions, say vi drawn independently from Fi. All players and auctioneer know the priors Fi, but only player i knows his draw vi. Definition Bayes-Nash Equilibrium: A set of strategies (s1, . . . sn) is a Bayes-Nash equilibrium for an auction if, for all i, agent i’s expected utility (taken over the random draws of v−i) is maximized by playing si(vi). When the BN equilibrium is si(vi) = vi, we say the auction is Bayes-Nash incentive compatible. Theorem The Revelation Principle: For any auction with a Bayes-Nash equlibrium there is an equivalent direct-revelation auction which results in same expected

• utcome and payments that is BN incentive compatible.

() November 28, 2011 7 / 23

BN equilibrium examples

First price auction, 3 players values uniform [0,1], s(v) = 2v/3. What do you need to check to verify this? All pay auction, 3 players values uniform [0,1], s(v) = 2v3/3.

() November 28, 2011 8 / 23

Characterization of Bayes-Nash Equilibrium

Fix an auction and assume each agent i uses strategy si. Let xi(vi) be the probability that agent i wins, taken over the random draws of v−i, when each agent j plays according to strategy sj. let pi(vi) be the expected payment of agent i, where the expectation is taken over the random draws of v−i. The set of strategies s = (s1, . . . , sn) is a Bayes-Nash equilibrium iff (Monotonicity:) xi(vi) is monotone non-decreasing in vi. (Payment Rule:) pi(vi) = vixi(vi) − ∞ xi(w)dw + pi(0). Corollary Revenue Equivalence: All auctions with the same allocation rule (in equilibrium) result in same auctioneer revenue.

() November 28, 2011 9 / 23

Revenue Equivalence

expected revenue of 1st price auction = expected revenue of 2nd price auction = expected revenue from all-pay auction. Useful for computing equilibrium strategies. Example: All pay auction, 3 players uniform [0,1]. Allocation rule is same as that of 2nd price auction.

() November 28, 2011 10 / 23

Profit Maximization

Assume agents’ values are drawn from known priors Fi. Suppose truthtelling is a BN equilibrium for some mechanism such that xi(vi) is the probability that agent i wins in equilibrium. By the payment identity we have pi(vi) = vixi(vi) − ∞ xi(w)dw. Therefore Evi(pi(vi)) = ∞

• vixi(vi) −

∞ xi(w)dw

• fi(vi)dvi.

Rearranging gives: Evi(pi(vi)) = ∞ xi

• vi − 1 − F(vi)

f(vi)

• fi(vi)dvi = Evi(xiψi(vi)),

where ψ(v) = v − 1 − F(v) f(v) .

() November 28, 2011 11 / 23

Myerson Mechanism

Evi(pi(vi)) = ∞ xi

• vi − 1 − F(vi)

f(vi)

• fi(vi)dvi = Evi(xiψi(vi)),

where ψ(v) = v − 1 − F(v) f(v) . Therefore, if we want to maximize profit, we should choose the feasible allocation that maximizes

• i

ψ(vi)xi. ψ(v) called v’s virtual value Potential Issue: This may not be truthful. This is a monotone allocation rule as long as ψi(vi) is monotone increasing in vi. The condition (on a probability distribution) that ψ(v) is monotone non-decreasing is called “regularity”.

() November 28, 2011 12 / 23

Profit Maximization with priors

For regular distributions, choosing the allocation that maximizes the sum

• f virtual values results in an allocation rule such that for each v−i,

xi(vi, v−i) is monotone non-decreasing in vi. Therefore, truthtelling is a dominant strategy! This mechanism maximizes expected auctioneer profit!

() November 28, 2011 13 / 23

Myerson Mechanism

Examples: two players, values drawn uniform [0,1] ψ(v) = v − 1 − F(v) f(v) = v − 1 − v 1 = 2v − 1, so ψ(v) ≥ 0 iff v ≥ 1/2. Resulting allocation rule: Allocation to highest bidder if value is at least 1/2 This is equivalent to Vickrey with reserve price of 1/2. Digital goods, n players values drawn iid from regular prior F. Let r be the value such that ψ(r) = 0. Profit maximizing auction: Vickrey with reserve price of r. Resulting payment rule: pay r if you win, 0 if lose. Note that payment rule slightly more complicated when agents values are drawn from different prior distributions.

() November 28, 2011 14 / 23

Profit Maximization without Prior

Example: digital goods auctions.

() November 28, 2011 15 / 23

Profit Maximization: No priors

How do we design truthful auctions that are (approximately) optimal in the worst-case in absence of prior?

1

Characterize Bayesian optimal mechanism OptF for every i.i.d. distribution F.

OptF is to charge each agent the price p that maximizes p(1 − F(p)).

2

Define distribution independent benchmark G(v) = sup

F

OptF(v).

Best fixed price auction!

3

Design a prior-free truthful mechanism that approximates G(v) on every v.

Random sampling auctions are constant competitive. (Deterministic auctions don’t work.)

Randomly partition bidders into two sets. Compute best fixed price profit in each part. Attempt to extract profit of each part from other part using truthful auction from homework.

() November 28, 2011 16 / 23

VCG

General technique for designing mechanisms for maximizing social welfare: Setting: Mechanism is selecting an outcome a from a set A of alternatives. Each player i has a value vi(a) for each outcome in A. Player utility is quasi-linear: If outcome a is selected, and player i required to pay pi his utility is vi(a) − pi. VCG: Choose outcome a∗ that maximizes

i vi(a)

payment of player j is

i=j vi(a′) − i=j vi(a∗), where a′ is the outcome

that maximizes

i=j vi(a′).

In other words, the payment is the harm that player’s presence causes to the rest of the world. This mechanism is truthful.

() November 28, 2011 17 / 23

Example: Matching Auctions

Setting: Collection of items Collection of buyers Each buyer i has a value vij for each item j.

• utput will be a matching.

Observations: VCG will choose as outcome the maximum weight matching, and sets the payments so as to incentivize truthful bidding. There is an iterative method that is guaranteed to converge to this

• utcome.

The prices are market clearing prices: all the items sell, each to someone who most wants them. Application: Ad Auctions (items are advertising slots, buyers are advertisers)

() November 28, 2011 18 / 23

Ad Auctions

The mechanism used in practice is the Generalized Second Price Auction: Rank bidders by product of bid and “quality”. Charge an advertiser the value of lowest bid with which he could win that slot. Properties of GSP: GSP is not truthful. GSP has a pure Nash equilibrium in which the allocation and payments are the same as those as VCG.

() November 28, 2011 19 / 23

Evolutionary Game Theory

A “mixed strategy” (p1, . . . , pn) is an evolutionary stable strategy if there is an x > 0 such that for all y ≤ x, and all i, if a fraction y of players play strategy i, and the rest play strategy p, the expected payoff of the players playing p is strictly higher than the expected payoff of the players playing i. Captures evoluationary stability of population frequencies. Every ESS is a NE, but not vice versa.

() November 28, 2011 20 / 23

Bargaining and Power in Networks

Question: how does a players’ “position” in a network of exchanges affect their outcome/profit? Nash bargaining solution: 2 players deciding how to split a $1, but player 1 (resp. 2) has an outside option of x (resp. y). If x + y ≤ 1, according to NBS, they will share the surplus (1 − x − y) equally. This is the equilibrium in a game in which they alternately make offers, with tiny probability p of breakdown at each step. Network Exchange: Given a network, with a player on each node, and a dollar on each edge, determine a matching (representing deals) and a split of dollar on each edge in matching.

Unstable outcome: A matching and values, such that 2 neighboring nodes are not matched, and the sum of their values is less than 1. Balanced outcome: A matching and values, such that for each edge in matching, split is Nash bargaining outcome, given best outside options for each node provided by values in rest of network.

() November 28, 2011 21 / 23

Price of Anarchy

Question: how much does selfishness hurt society? Congestion games: Each player choosing a subset of resources from among a collection of possible resources. Congestion/latency on a resource is a function of the number of players choosing it. A player’s total latency is sum of latencies on resources it selects (which depends

• n choices of other players).

Such games are “potential games”: They have a pure NE that can be reached by best-response dynamics. Braess paradox: Adding capacity can hurt average latency in Nash equilibrium. Price of anarchy: worst-case ratio between average latency in NE and average latency in socially optimal outcome. For routing (each player chooses a path from a source to a destination), with linear latency functions, the price of anarchy is at most 4/3.

() November 28, 2011 22 / 23

Voting Theory

Assume at least 3 alternatives: Combining individual rankings of alternatives = ⇒ combined ranking:

Condorcet Paradox: Using majority on each pair of candidates, for arbitrary preferences, not transitive. If preferences are single-peaked (one maximum), majority works. Arrow’s impossibility theorem: Any reasonable voting scheme (reasonable = produces transitive ranking + unanimity + independence of irrelevant alternatives) is a dictatorship.

Combining individual rankings of alternative = ⇒ winner:

Gibbard/Satterthwaite: Any voting rule that is not a dictatorship and in which any candidate can win will create an incentive for strategic voting/manipulation in some circumstances. Together Majority rule and Rank order voting dominate all other voting rules.

() November 28, 2011 23 / 23