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  1. Cooperative Games, Mechanism Design, and Auctions Onn Shehory March 9-13 2009 Politecnico di Milano

  2. The Glove Game • Players set: {1,2,3} • 1,2 have right-had gloves • 3 has left-had gloves • Players may stay alone and profit 0 • Or join together: {1,3}, {2,3}, {1,2,3} to gain 1 • What mechanism should they follow? • How do they behave given such mechanism? • How is profit divided?

  3. The English Auction • An item is placed for sale • Players free to bid • New bid must be higher than current • Winner: highest bid

  4. Outline • Protocols and strategies • Attitudes and rationality • Stability and equilibrium • Nash revisited • Cooperative games • Solution concepts • Coalition Formation • RFP coalitions • Mechanism design • Auctions • Auctions – field results

  5. What is a Mechanism/Protocol? • A protocol (aka mechanism): • provides a set of rules and behaviors to be followed by its participants • following the rules of a protocol is to a player’s discretion, though deviation may leave her “out of the game” • examples: auctions, negotiation, voting • Desired properties: • maximize payoffs • not manipulable/enforceable • simple to implement and execute

  6. What are Strategies? • A strategy: • is one of the possible actions a player can select given the protocol • is not dictated (or provided) by the protocol • is usually the result of the player’s reasoning and decisions, based on local algorithms and information • examples: • in an auction – bid as low as possible • in elections – vote for your faction • A good strategy: • should maximize the player’s payoff given the protocol and the behavior of other players • should be difficult or impossible to manipulate • should be computationally feasible • may depend on the strategies of other players

  7. Players Attitudes • Self-interest: a self-interested player is attempting to maximize its own personal payoff • Benevolence/altruism: a benevolent player is attempting to increase others’ payoffs and the cumulative payoff of the society

  8. Risk Attitudes • Risk prone: a player that has a preference for risk • Risk averse: a player that has a preference for avoiding risk • Risk neutral: a player that has no risk preference • Players do not need to be strictly prone, neutral or averse – they may mix these • Human players tend to have alternating and context dependent risk attitudes

  9. Example Risk Attitudes • You are offered two options: • Get 1000 Euro, cash • Get a lottery certificate, with a prize value of 10,000 Euro, and a 10% chance of winning • What should you choose? • Select 2, for a chance for getting 10,000? • Select 1 to avoid risk? • What should you choose if chance is 20%? • Select 2, to maximize expected utility (2000)?

  10. Rationality • A rational behavior is such that prefers a greater payoff over a smaller one • A rational player should always behave rationally. That is, from among several options available, he should select the one that results in maximum payoff • The problem: • in may cases the number of options is overwhelming • there may be no algorithm for finding the best

  11. Bounded Rationality • To overcome problems of rationality, bounded rationality: • limits the time/computation for option consideration • prunes the search space • imposes restrictions on the types of options • Results in fewer possibilities, hence • computationally feasible • may be too restrictive, far from optimal • strategically inferior to rational

  12. “Good Enough” Behavior • Make the bounded rationality rational: • modify linear payoff functions to incorporate computational costs • put a cap on payoff • add a small-amounts’ indifference • The payoff of an option is good enough if • too much additional computation to find other good options, or • other options do not provide a significant payoff increase, or • the player is indifferent w.r.t. the increase

  13. Stability and Equilibria

  14. Protocol Evaluation • Payoff maximization: can refer to individual payoffs, group payoffs, or social welfare - the sum of individual payoffs • Pareto-optimality: a payoff vector p(x1,x2,…,xn) is Pareto-optimal if there is no other feasible payoff vector p' such that at least one payoff is better in p' and no payoff is worse in p • Stability: a protocol is stable if once the players arrived at a solution they do not deviate from it

  15. Stability and Equilibria • There are multiple stability concepts. In game theory, the notion of equilibrium is used: • dominant strategies: the agents have some strategies that, regardless of what others do, maximize payoff • Nash equilibrium: the agents have strategies that, as long as other stick to theirs, maximize payoff • Mixed Nash: the agents each have a set of strategies from among which they select one with some probability • Bayes-Nash: adds types (e.g. history) to the previous one

  16. The Prisoner’s Dilemma

  17. No Pure Nash Equilibrium

  18. No Pure Nash Equilibrium

  19. Mixed Nash • Player 1 will cooperate with probability pc and defect with probability pd • Player 2 will cooperate with probability qc and defect with probability qd • Expected utility of an agent is the utility from a strategy times the probability of this strategy being selected • When there are multiple possibilities, the expected utility is a sum over these possibilities

  20. Computing the Probabilities • The expected utility of an player x when the other player y follows strategy s is denoted by Ux(s) • In the case of equilibrium (mixed Nash), the expected utility of x should be the same for all of the possible strategies of y • In our case we have players 1,2 and strategies c,d • We require that Ux(c) = Ux(d), which means that for each of the two players, the expected utility from the other cooperating should be equal to the expected utility from the other defecting

  21. Computation Details • For player 1 we have: • U1(c)=6 pc+ 0 pd, U1(d)=0 pc+ 5 pd • For player 2 we have: • U2(c)=2 qc+ 3 qd, U2(d)=1 qc+ 0 qd • The requirement that Ux(c) = Ux(d) results in: • qc= 0.642, qd = 0.358 • pc= 0.317, pd = 0.683

  22. Tragedy of Common Goods • Information on the web is (mostly) free • Web agents that seek up to date information may query web site as frequently as desired • If all agents will do so, the network will be overly congested, and some servers will crash • So, is it undesirable to behave this way? • If all (or most) of the agents prevent congestion, it is in the best interest of each individual agent to increase network use ...

  23. Cooperative Games

  24. Cooperative Games • A cooperative game (aka coalitional game): • Cooperation within groups is enforceable • Groups compete, and not individuals • Each group (=coalition) has a value v • Characteristic function: v : 2N, from coalitions to payments • Players decide which coalitions to form (to maximize payoff)

  25. Coalitional Games • Coalitional game (N,v) • A set of players N • A coalitionS is a group of players, subset of N, which cooperate • Value (or utility) of a coalition v • v(S)is a real, represents the gain of coalition S in the game (N,v) • v(N)is the value of forming the grand coalition, coalition of all players • Player payoff xi • The portion of v(S) received by a player i in coalition S • Characteristic function form implies: • vdepends only on the internal structure of the coalition • Transferable utility • The value of a coalition can be distributed arbitrarily among its players

  26. Coalitional Games: Example • Example: Majority Vote • Prime minister is elected by majority vote • A coalition consisting of a majority of players has a worth of 1 since it is a decision maker • Value of a coalition does not depend on the external strategies of the users => characteristic function form

  27. Super-additive Games • Super-additive game • (N,v)is super-additive if • Here, cooperation is always beneficial • Unification of two coalitions increases overall payoff • Monotonicity: larger coalitions gain more • Not all games are super-additive!

  28. Coalitional Stability • Stability of a coalition • Depends on how the value visdistributed among the players • How to divide v? • Improper payoff division => players may leave the coalition, unhappy with their share • Multiple solution concepts address this point

  29. Coordination Game • For 1, A>C, D>B • For 2, a>c, d>b • Red circles are pure Nash • For driving side, all benefit if all adopt the same side, but two equilibria points exist

  30. Solution Concepts

  31. Solution Concepts • Assumption: the grand coalition will form • Even when the solution includes multiple coalitions • Solve for sub-games • Each players gets a payoff xi • Major issue: payoff distribution • A solution concept is a payoff allocation vector x N • Efficiency: ∑ xi = v(N) • Individual rationality: xi ≥ v(i) • Group rationality: v(S)≤ ∑ xi S • Imputation: a payoff allocation vector which is efficient and individually rational (and group rational for N) • Many solution concepts are imputations • Players prefer coalitions based on their respective payoffs

  32. Example • Socks selling • Sold in pairs for 3euro a pair • 2 sellers, each holds 5 socks • Each can get 6euro, but together they get 15euro. • Imputations: (6, 9), (7, 8), (7.5, 7.5)

  33. Some Properties • Null player: when added to a coalition, contributes nothing to its value • Existence (of a solution concept): determines whether the solution concept exists for every game • Examples: Kernel exists, Core does not • Symmetry: symmetric players receive equal payoffs • Uniqueness: the solution concept is unique

  34. The Shapley Value (1953) • Unique, efficient, symmetric, allocates zero to null players • Individually rational for super-additive games • The payoff allocated to player i is φi(v) = 1/n! ΣSN\i (s!-(n-s-1)!) (v(S∪i)-v(S)) • s = |S| • This is considered a fair allocation

  35. The Glove Game Example • N={1,2,3} • 1,2 have right-had gloves • 3 has left-had gloves • v(S) = 1 for {1,3}, {2,3}, {1,2,3}, otherwise 0 • φ1(v) = 1/6 • φ2(v) = 1/6 • φ3(v) = 4/6

  36. The Stable Setvon Neumann & Morgenstern (1944) • Given a game v and imputations x,y • x dominates y if • there is a nonempty coalition S, such that • the members of S prefer payoff from x over payoff from y • v(S) ≥ ∑ xi S • S players can threaten to quit the grand coalition if x not implemented

  37. Stable Set Example • x=(1,2,3), y=(3,2,1) • For players 1,2, y is better than x • Does y dominate x? • Depends on v({1,2}) • If v({1,2}) ≥ 5, y dominates x via {1,2}

  38. Stable Set Example 2 • x=(50,50,0), y=(0,60,40), z=(15,0,85) • v({1,2}) = v({1,3}) = v({2,3}) =100 • y dominates x via {2,3} • z dominates y via {1,3} • x dominates z via {1,2}

  39. Some Properties • Existence: the stable set may or may not exist • Uniqueness: when exists, it is usually not unique • Internal stability: no imputation within the stable set dominate one another • External stability: all payoff vectors outside the stable set are dominated by at least on member of the set • Interpretation: the stable set represents conflicts, but excludes inferior behaviors

  40. The Core • The coreis a set of imputations(x1, . . .,xN)satisfying two conditions • No coalition has a value greater than the sum of its members’ payoffs (coalition rationality) • The core can be empty • A non-empty core in a super-additive game => stable grand coalition • No coalition has an incentive to leave (and receive a greater payoff)

  41. Properties of the Core • May be empty (rather common) • Not unique • Subset of the stable set • Has several variants • E.g., epsilon-core • Least-core

  42. Shoes Example • A pair: a left and a right shoe. Sold for €50 • 21 players: 10 have 1 left shoe, 11 have 1 right shoe • The core: a single imputation, gives 10 to left shoe owners, 0 to right shoe owners • Any left-right pair can form a coalition and sell for €50 • Any such pair getting less than that will block the imputation • For imputation in the core, any of these pairs gets exactly 50 (we can only sell 10 pairs, totaling to 500) • One right-shoe owner gets 0 payment • Examine the pairs: if any left-shoe owner gets less than 50, say 40, it can join this player, sell their shoes, give her 5, and keep 45 to herself. Both are better off • But such a left-shoe owner cannot exist: all left shoe owners get already 50. • This holds for any unequal partition • The core is very sensitive to oversupply.

  43. The KernelDavis & Maschler (1965) • We do not assume imputations • That’s it – no group rationality • Efficiency still required • Result: we are interested no only in payoff distribution, but in the coalitions that form • Implicitly assumes bargaining (but in practice computes its result)

  44. Surplus • Maximum surplus • Given an efficient payoff vector x and players i,j, the maximum surplus of i over j is: sij(x)= max(v(S)- ∑kSxk : iS, jS) • the maximal amount i can gain without the cooperation of j by withdrawing from the grand coalition N (where x applies), other players in i's withdrawing coalition are satisfied with their x payoffs • The maximum surplus measures a player's bargaining power over another

  45. The Kernel Defined • The kernel is the set of payoff vectors x that satisfy, for all pairs i,j: (sij(x)-sji(x))·(xj - v(j)) ≤ 0 and (sji(x)-sij(x))·(xi - v(i)) ≤ 0 • If sij(x) > sji(x) then ioutweighsj – it has more bargaining power • But ifxj = v(j), j can obtain xj on his own, thus i’s threat is invalidated • For vectors in the kernel, no player can outweigh another – no valid threat  stability

  46. Kernel Properties • Equilibrium of agents‘ surpluses: In each coalition no player can outweigh another, thus getting a better payoff (surplus) in an alternative coalition excluding the opponent „I can get more without you, than you can without me.“ • Exists, Pareto-optimal, not unique • A subset of the bargaining set • Exponentially hard to compute • Computational solution: Stearns (1968) • May converge very slowly • A single point out of many • Polynomial variants (Shehory/Kraus 1996; Klusch/Shehory 1996)

  47. Example • Three players • v(i)=0, v(12)=2, v(13)=3, v(23)=4, v(123)=8 • Kernel: 2, 2.5, 3.5 • Shapley: 13/6, 16/6, 19/6 • v(i)=0, v(12)=2, v(13)=3, v(23)=4, v(123)=5 • Kernel: 2/3, 5/3, 8/3 • Shapley: 7/6, 10/6, 13/6

  48. More Concepts • The Bargaining set • The Nucleolus • Both based on excess: v(S)- ∑kSxk • Possible gain of players when quitting a coalition

  49. Coalition Formation

  50. Coalitions and Complexity • Given N players, there are 2N-1 different possible coalitions • If there are k tasks, may need to multiply by exp(k) • The number of configurations is O(N(N/2)) • Hence, exhaustive search is infeasible • Additionally, players may have conflicting preferences over the possible configurations • Nevertheless, coalitions are beneficial