Mechanism Design II. Terence Johnson. University of Notre Dame. Terence Johnson (ND) Mechanism Design II 1 / 30

Transcription

1 Mechanism Design II Terence Johnson University of Notre Dame Terence Johnson (ND) Mechanism Design II 1 / 30

2 Mechanism Design Recall: game theory takes the players/actions/payoffs as given, and makes predictions about how people behave, while mechanism design takes the players and their preferences as given, and designs actions/games to get specific outcomes Terence Johnson (ND) Mechanism Design II 2 / 30

3 Mechanism Design Recall: game theory takes the players/actions/payoffs as given, and makes predictions about how people behave, while mechanism design takes the players and their preferences as given, and designs actions/games to get specific outcomes Today, we learn the core concepts for a much more general theory of incentive design Terence Johnson (ND) Mechanism Design II 2 / 30

4 The Revelation Principle The key tool in mechanism design is the Revelation Principle: every equilibrium in every game where the agents have private information can be thought of as a different game, in which the agents make a report (not necessarily truthfully) about their private information to the mechanism or the referee, and then the mechanism or referee plays the game for them as they would have played Terence Johnson (ND) Mechanism Design II 3 / 30

5 The Revelation Principle The key tool in mechanism design is the Revelation Principle: every equilibrium in every game where the agents have private information can be thought of as a different game, in which the agents make a report (not necessarily truthfully) about their private information to the mechanism or the referee, and then the mechanism or referee plays the game for them as they would have played Terence Johnson (ND) Mechanism Design II 3 / 30

6 The Model There are i = 1,..., N agents Terence Johnson (ND) Mechanism Design II 4 / 30

7 The Model There are i = 1,..., N agents Each player i has a privately known type, θ i Terence Johnson (ND) Mechanism Design II 4 / 30

8 The Model There are i = 1,..., N agents Each player i has a privately known type, θ i There is an outcome, x = (x 1, x 2,..., x N ) Terence Johnson (ND) Mechanism Design II 4 / 30

9 The Model There are i = 1,..., N agents Each player i has a privately known type, θ i There is an outcome, x = (x 1, x 2,..., x N ) Each player s payoff is x i θ i t i, and t i is a payment Terence Johnson (ND) Mechanism Design II 4 / 30

10 Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i Terence Johnson (ND) Mechanism Design II 5 / 30

11 Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). Terence Johnson (ND) Mechanism Design II 5 / 30

12 Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments Agent i s payoff is x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). U i (ˆθ i, ˆθ i, θ i ) = x i (ˆθ i, ˆθ i )θ i t i (ˆθ i, ˆθ i ) Terence Johnson (ND) Mechanism Design II 5 / 30

13 Direct Mechanisms While θ i is agent i s true type, agent i can always lie and report any alternative type, ˆθ i A direct mechanism prescribes an allocation and a set of payments Agent i s payoff is x(ˆθ) = (x 1 (ˆθ), x 2 (ˆθ),..., x N (ˆθ)) t(ˆθ) = (t 1 (ˆθ), t 2 (ˆθ),..., t N (ˆθ)). U i (ˆθ i, ˆθ i, θ i ) = x i (ˆθ i, ˆθ i )θ i t i (ˆθ i, ˆθ i ) This lets us talk about how what outcomes (x, t) can be achieved, without worrying about the actual game (first or second price auction? TTC or SRD? etc) Terence Johnson (ND) Mechanism Design II 5 / 30

14 The SPAR as a direct mechanism For the second price auction, the direct mechanism is { 1, ˆθ i > max j i ˆθ j x i (ˆθ i, ˆθ i ) = 0, otherwise and t i (ˆθ i, ˆθ i ) = { max j i ˆθj, ˆθi > max j i ˆθj 0, otherwise Terence Johnson (ND) Mechanism Design II 6 / 30

15 Vickrey Auctions: Dominant Strategy Implementation The Vickrey-Clarke-Groves mechanism or VCG mechanism is: and the payment for each i, x (ˆθ) = max x I v i (x, ˆθ i ) i=1 t i (ˆθ) = W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) Terence Johnson (ND) Mechanism Design II 7 / 30

16 Vickrey Auctions: Dominant Strategy Implementation The Vickrey-Clarke-Groves mechanism or VCG mechanism is: and the payment for each i, x (ˆθ) = max x I v i (x, ˆθ i ) i=1 t i (ˆθ) = W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) We take the efficient action given the reports, and charge each person who participates the difference between gross welfare of the other agents without their report and gross welfare of the other agents given their report Terence Johnson (ND) Mechanism Design II 7 / 30

17 Vickrey Auctions: Dominant Strategy Implementation The Vickrey-Clarke-Groves mechanism or VCG mechanism is: and the payment for each i, x (ˆθ) = max x I v i (x, ˆθ i ) i=1 t i (ˆθ) = W \i (, ˆθ i ) W i (ˆθ i, ˆθ i ) We take the efficient action given the reports, and charge each person who participates the difference between gross welfare of the other agents without their report and gross welfare of the other agents given their report We are internalizing the externality by charging each participant their impact on gross welfare this implements the efficient outcome and gives agents a weakly dominant strategy to report honestly Terence Johnson (ND) Mechanism Design II 7 / 30

18 Selling a unit of a good To keep things simple, we imagine a seller designing a market to sell a single unit of a good, so that the allocation x i (ˆθ) gives the probability that i receives the good, as a function of the reports, and θ i is i s valuation for the good. Terence Johnson (ND) Mechanism Design II 8 / 30

19 The Bayesian Framework We assume the agents and design have the same beliefs: agent i s type θ i is independent of the other agents types, according to a distribution F i (θ i ) with density f i (θ i ) on some range [0, θ] Then the direct utility function is ] U i (ˆθ i, θ i ) = [x Eˆθ i i (ˆθ i, ˆθ i )θ i t i (ˆθ i, ˆθ i ) So we are explicitly allowing the agents and seller to have beliefs (F 1, F 2,..., F I ) about the agents types Terence Johnson (ND) Mechanism Design II 9 / 30

20 Individual Rationality A direct mechanism (x, t) is individually rational if for every buyer i and every type θ i, U i (θ i, θ i ) 0, so that honest reporting yields at least a payoff of zero, the agent s outside option Terence Johnson (ND) Mechanism Design II 10 / 30

21 Incentive Compatibility A direct mechanism (x, t) is incentive compatible if for every buyer i and every type θ i and every alternative type θ i, U i (θ i, θ i ) U i (θ i, θ i ), so that honest reporting is at least as good as lying and reporting something else Terence Johnson (ND) Mechanism Design II 11 / 30

22 Myerson s Theorem Theorem Consider a mechanism (x, t) in which r is the lowest type that trades with any positive probability. (x, t) is incentive compatible if and only if ] 1 E θ i [x i (ˆθ i, θ i ) is non-decreasing in ˆθ i 2 E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )θ i ] θ i r x i (z, θ i )dz If the worst-off type gets a payoff U i (r, r) = 0, the mechanism is individually rational. Terence Johnson (ND) Mechanism Design II 12 / 30

23 Myerson s Theorem Theorem Consider a mechanism (x, t) in which r is the lowest type that trades with any positive probability. (x, t) is incentive compatible if and only if ] 1 E θ i [x i (ˆθ i, θ i ) is non-decreasing in ˆθ i 2 E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )θ i ] θ i r x i (z, θ i )dz If the worst-off type gets a payoff U i (r, r) = 0, the mechanism is individually rational. So for a mechanism to be incentive compatible, reporting a higher type results in a (weakly) higher likelihood of winning. Then the expected payment is pinned down by how the good is awarded as a function of the reports. Terence Johnson (ND) Mechanism Design II 12 / 30

24 Suppose a mechanism is incentive compatible. Take two incentive compatibility constraints, U i (θ i, θ i ) U i (θ i, θ i) and U i (θ i, θ i ) U i(θ i, θ i ) and assume θ i > θ i. This implies [ E θ i [x i (θ i, θ i )θ i t i (θ i, θ i )] E θ i xi (θ i, θ i )θ i t i (θ i, θ i ) ] E θ i [ xi (θ i, θ i )θ i t i (θ i, θ i ) ] E θ i [ xi (θ i, θ i )θ i t i (θ i, θ i ) ]. Terence Johnson (ND) Mechanism Design II 13 / 30

25 Suppose a mechanism is incentive compatible. Take two incentive compatibility constraints, U i (θ i, θ i ) U i (θ i, θ i) and U i (θ i, θ i ) U i(θ i, θ i ) and assume θ i > θ i. This implies [ E θ i [x i (θ i, θ i )θ i t i (θ i, θ i )] E θ i xi (θ i, θ i )θ i t i (θ i, θ i ) ] E θ i [ xi (θ i, θ i )θ i t i (θ i, θ i ) ] E θ i [ xi (θ i, θ i )θ i t i (θ i, θ i ) ]. Add these; the transfers cancel out, leaving E θ i [x i (θ i, θ i )] θ i + E θ i [ xi (θ i, θ i ) ] θ i E θ i [ xi (θ i, θ i ) ] θ i + E θ i [x i (θ i, θ i )] θ i (1) Rearrange this to get (θ i θ i) ( [ Eθ i [x }{{} i (θ i, θ i )] E θ i xi (θ i, θ i ) ]) 0 >0 So that if θ i > θ i and IC, then E θ i [x i (θ i, θ i )] E θ i [x i (θ i, θ i)], which is (i). Terence Johnson (ND) Mechanism Design II 13 / 30

26 Now consider U i (θ i, θ i ) = E θ i [x i (θ i, θ i )] θ i E θ i [t i (θ i, θ i )]. Re-arranging this yields E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )] θ i U i (θ i, θ i ). Terence Johnson (ND) Mechanism Design II 14 / 30

27 Now consider Re-arranging this yields U i (θ i, θ i ) = E θ i [x i (θ i, θ i )] θ i E θ i [t i (θ i, θ i )]. E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )] θ i U i (θ i, θ i ). If we can solve for U i (θ i, θ i ), then we know what the payment must be. But incentive compatibility implies that honest reporting is payoff-maximizing. Terence Johnson (ND) Mechanism Design II 14 / 30

28 Now consider Re-arranging this yields U i (θ i, θ i ) = E θ i [x i (θ i, θ i )] θ i E θ i [t i (θ i, θ i )]. E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )] θ i U i (θ i, θ i ). If we can solve for U i (θ i, θ i ), then we know what the payment must be. But incentive compatibility implies that honest reporting is payoff-maximizing. Consider du i (θ i, θ i ) dθ i = d [ ] U i (ˆθ i, θ i ) +E θ i [x i (θ i, θ i )], } d ˆθ {{} FONC=0 and U i (θ i, θ i ) = θi r E θ i [x i (z, θ i )] dz + U i (r, r). }{{} =0 Terence Johnson (ND) Mechanism Design II 14 / 30

29 Then we can substitute that in, yielding θi E θ i [t i (θ i, θ i )] = E θ i [x i (θ i, θ i )] θ i E θ i [x i (z, θ i )] dz, r which is (ii). Terence Johnson (ND) Mechanism Design II 15 / 30

30 Now suppose i and ii; we ll show that (x, t) is incentive compatible. Consider U i (θ i, θ i ) U i (θ i, θ i ) = U i (θ i, θ i ) U i (θ i, θ i) + U i (θ i, θ i) U i (θ i, θ i ) = = θi θ i θi where the first line follows from (ii). θ i E θ i [x i (z, θ i )] dz θi θ i E θ i [ xi (θ i, θ i ) ] dz E θ i [x i (z, θ i )] E θ i [ xi (θ i, θ i ) ] dz. Terence Johnson (ND) Mechanism Design II 16 / 30

31 Now suppose i and ii; we ll show that (x, t) is incentive compatible. Consider U i (θ i, θ i ) U i (θ i, θ i ) = U i (θ i, θ i ) U i (θ i, θ i) + U i (θ i, θ i) U i (θ i, θ i ) = = θi θ i θi θ i E θ i [x i (z, θ i )] dz θi θ i E θ i [ xi (θ i, θ i ) ] dz E θ i [x i (z, θ i )] E θ i [ xi (θ i, θ i ) ] dz. where the first line follows from (ii). Now, the integrand has the same sign as the domain of integration [θ i, θ i], since the probability of winning is non-decreasing in the report: it is always positive. Therefore U i (θ i, θ i ) U i (θ i, θ i), and (x, t) is incentive compatible. Terence Johnson (ND) Mechanism Design II 16 / 30

32 The FPAR is incentive compatible In the FPAR, the probability of winning is the probability of submitting the highest bid, which is the probability of having the highest type: raising your bid is equivalent to saying you have a higher type, so it satisfies i. Terence Johnson (ND) Mechanism Design II 17 / 30

33 The FPAR is incentive compatible In the FPAR, the probability of winning is the probability of submitting the highest bid, which is the probability of having the highest type: raising your bid is equivalent to saying you have a higher type, so it satisfies i. In the FPAR, U i (θ i, θ i ) = E θ i [ xi (θ i, θ i )θ i x i (θ i, θ i )t i (θ i, θ i )) ] and D θi U i (θ i, θ i ) = E θ i [x i (θ i, θ i )] so it satisfies ii. Terence Johnson (ND) Mechanism Design II 17 / 30

34 The SPAR is incentive compatible In the SPAR, the probability of winning is the probability of reporting the highest bid, which is the probability of having the highest type: raising your bid is equivalent to saying you have a higher type, so it satisfies i. Terence Johnson (ND) Mechanism Design II 18 / 30

35 The SPAR is incentive compatible In the SPAR, the probability of winning is the probability of reporting the highest bid, which is the probability of having the highest type: raising your bid is equivalent to saying you have a higher type, so it satisfies i. In the SPAR, U i (θ i, θ i ) = E θ i [ xi (θ i, θ i )θ i x i (θ i, θ i )θ (2) ] and D θi U i (θ i, θ i ) = E θ i [x i (θ i, θ i )] so it satisfies ii. Terence Johnson (ND) Mechanism Design II 18 / 30

36 Myerson s Theorem This is a helpful tool: it tells us when a mechanism is incentive compatible as a consequence of its quantitative properties, instead of a bunch of inequalities Terence Johnson (ND) Mechanism Design II 19 / 30

37 Myerson s Theorem This is a helpful tool: it tells us when a mechanism is incentive compatible as a consequence of its quantitative properties, instead of a bunch of inequalities This can be used for Evil. I will now show you the Evil. Terence Johnson (ND) Mechanism Design II 19 / 30

38 Profit Maximization Suppose the designer wishes to do [ I ] max (x,t) E θ i=1 t i (θ) subject to incentive compatibility and individual rationality. Terence Johnson (ND) Mechanism Design II 20 / 30

39 Profit Maximization Suppose the designer wishes to do [ I ] max (x,t) E θ i=1 t i (θ) subject to incentive compatibility and individual rationality. We have E θ i [t i (θ i, θ i )]. We need E θi [E θ i [t i (θ i, θ i )]], which is θ 0 f i (θ i )E θ i [t i (θ i, θ i )]dθ i Terence Johnson (ND) Mechanism Design II 20 / 30

40 Integration by Parts We need a tool you probably forgot The product rule is ((1 F (x))g(x)) = f (x)g(x) + (1 F (x))g (x), and the inverse of the product rule is θ f (x)g(x)dx = r θ This is called integration by parts r (1 F (x))g (x)dx + [(1 F (x))g(x)] θ r Terence Johnson (ND) Mechanism Design II 21 / 30

41 Profit maximization We need to compute the expected payment for i, θ r θi f (θ i )E θ i [x i (θ i, θ i )]θ i f (θ i ) E θ i [x i (z, θ i )]dz dθ i } r {{ } Problem term Terence Johnson (ND) Mechanism Design II 22 / 30

42 Profit maximization We need to compute the expected payment for i, θ r θi f (θ i )E θ i [x i (θ i, θ i )]θ i f (θ i ) E θ i [x i (z, θ i )]dz dθ i } r {{ } Problem term The problem term can be isolated as θ r which integrates by parts to [ θi f (θ i ) r ] E θ i [x i (z, θ i )]dz } {{ } G(θ i ) dθ i θi r (1 F i (θ i ))E θ i [x i (θ i, θ i )]dθ i + 0 Terence Johnson (ND) Mechanism Design II 22 / 30

43 Virtual Types Then i s expected payment is or We call θi r [{ f (θ i )E θ i θ i 1 F } ] i(θ i ) x i (θ i, θ i ) dz, f i (θ i ) [{ E θ [t i (θ)] = E θ θ i 1 F } ] i(θ i ) x i (θ i, θ i ). f i (θ i ) ψ i (θ i ) = θ i 1 F i(θ i ) f i (θ i ) the virtual type: it is an informationally adjusted measure of the agent s willingness to pay Terence Johnson (ND) Mechanism Design II 23 / 30

44 Virtual Types Where have you seen the virtual type before? Consider the monopolist with cost c facing agent i: with FONC which can be re-arranged max (1 F i (t i ))(t t i }{{} i c) Demand (1 F i (t i )) f i (t i )(t i c) = 0 t i 1 F i(t i ) f i (t i ) }{{} c = 0 }{{} MC MR So ψ i (t i ) is the marginal revenue associated with selling to the t i type. Terence Johnson (ND) Mechanism Design II 24 / 30

45 Virtual Types Uniformly distributed types: F (θ i ) = θ i on [0, 1]. Then ψ(θ i ) = θ i 1 θ i 1 = 2θ i 1 Exponentially distributed types: F (θ i ) = 1 e λθ i on [0, ). Then ψ(θ i ) = θ i e λθ i λe λθ i = θ i 1 λ For the Normal distribution, you can t get a closed form solution: ψ(θ i ) = θ i θ i 1 2πσ 2 e (z µ)2 /(2σ 2) dz 1 2πσ 2 e (θ i µ) 2 /(2σ 2 ) Terence Johnson (ND) Mechanism Design II 25 / 30

46 Profit maximization This means that [ I ] [ I ] E θ t i (θ) = E θ x i (θ)ψ i (θ i ). i=1 i=1 Notice this only depends on the x, not the t: we re just deciding the winner here. Terence Johnson (ND) Mechanism Design II 26 / 30

47 Profit maximization This means that [ I ] [ I ] E θ t i (θ) = E θ x i (θ)ψ i (θ i ). i=1 i=1 Notice this only depends on the x, not the t: we re just deciding the winner here. The optimal allocation of the good is { xi 1, ψ i (θ i ) > max j i ψ j (θ j ), ψ i (θ i ) 0 (θ) = 0, otherwise. Terence Johnson (ND) Mechanism Design II 26 / 30

48 Profit maximization This means that [ I ] [ I ] E θ t i (θ) = E θ x i (θ)ψ i (θ i ). i=1 i=1 Notice this only depends on the x, not the t: we re just deciding the winner here. The optimal allocation of the good is { xi 1, ψ i (θ i ) > max j i ψ j (θ j ), ψ i (θ i ) 0 (θ) = 0, otherwise. If an agent s type θ i yields a negative virtual value, it s not profitable to sell to them, defining the worst-off type ri for agent i as r i 1 F i(ri ) f i (ri ) = 0. (Notice, this is the optimal reserve price we found earlier). Terence Johnson (ND) Mechanism Design II 26 / 30

49 Revenue equivalence Notice that expected revenue, [ I ] [ I ] E θ t i (θ) = E θ x i (θ)ψ i (θ i ). i=1 i=1 only really depends on x(θ). Terence Johnson (ND) Mechanism Design II 27 / 30

50 Revenue equivalence Notice that expected revenue, [ I ] [ I ] E θ t i (θ) = E θ x i (θ)ψ i (θ i ). i=1 i=1 only really depends on x(θ). Theorem If two incentive compatible mechanisms (x, t) and (x, t ) give agents the same expected probability of winning, then the two mechanisms raise the same amount of revenue. Terence Johnson (ND) Mechanism Design II 27 / 30

51 The FPAR and SPAR Suppose the agents types are identically distributed, so that F i (θ i ) = F (θ i ) for all i Terence Johnson (ND) Mechanism Design II 28 / 30

52 The FPAR and SPAR Suppose the agents types are identically distributed, so that F i (θ i ) = F (θ i ) for all i Then they all have the same virtual types, ψ(θ i ) = θ i 1 F (θ i) f (θ i ) Terence Johnson (ND) Mechanism Design II 28 / 30

53 The FPAR and SPAR Suppose the agents types are identically distributed, so that F i (θ i ) = F (θ i ) for all i Then they all have the same virtual types, ψ(θ i ) = θ i 1 F (θ i) f (θ i ) Then the agent with the highest type is also the agent with the highest virtual type the FPAR and SPAR with reserve prices satisfying r 1 F (r ) f (r = 0 ) are both optimal among all possible ways of selling the goods. Terence Johnson (ND) Mechanism Design II 28 / 30

54 When are FPAR/SPAR/HRB/LRB-type auctions optimal? The supply of goods is fixed or marginal costs are strictly increasing The agents types are identically distributed Terence Johnson (ND) Mechanism Design II 29 / 30

55 Conclusion I use these tools to design cross-subsidization programs to help poor households afford sanitation services, and to select the lowest-cost suppliers in procurement auctions when there s a matching issue between suppliers and jobs Basically, we estimate virtual types conditional on observables using an experiment, then re-design the market to function more efficiently So now you have two approaches: Vickrey to maximize welfare, the Virtual Type one to maximize profits, and they can be combined to solves all kinds of other problems Next time we ll talk about how to estimate F (θ i ) from data Terence Johnson (ND) Mechanism Design II 30 / 30