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Probability Independent Equilibrium: Combinatorial Auctions

This paper discusses games without probabilistic information (PRI) and games with probabilistic information (Bayesian games) in the context of combinatorial auctions. It explores dominance in games without PRI and equilibrium concepts such as PI Equilibrium and Bayesian Equilibrium. The Generalized Vickrey Auction, Vickrey-Clarke-Groves Mechanism, and Groves Mechanism are also discussed.

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Probability Independent Equilibrium: Combinatorial Auctions

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  1. Probability Independent Equilibrium:Combinatorial Auctions Noa Kfir-Dahav Dov Monderer Moshe Tennenholtz June2001

  2. a1211 b1211 a2211 b2211 a1111 b1111 a2111 b2111 a1212 b1212 a2112 b2112 a2212 b2212 a1112 b1112 a2221b2221 a2121b2121 a1121b1121 a1221b1121 a1222 b1222 a2222 b2222 a1122 b1122 a2122 b2122 s1 s2 Games without probabilistic information t1 t2

  3. Games without PRI (ii) b1211=b(t1,s2, Row 1, Col. 1) In a more general case, the payoff of agent i depends on a vector of types, one for each agent, and on a vector of actions, one for each agent: ui=ui(ti,t-i, xi,x-i) In the most general case Nature also has a type t0.

  4. a2211 b2211 a2111 b2111 a1111 b1111 a1211 b1211 a1212 b1212 a2212 b2212 a2112 b2112 a1112 b1112 a1121b1121 a1221b1121 a2221b2221 a2121b2121 a2122 b2122 a2222 b2222 a1222 b1222 a1122 b1122 s1 s2 Games with probabilistic information (Bayesian Games) t1 p11 p12 t2 p21 p22

  5. 1, 1 5, 0 5, 0 1, 1 0, 5 4, 4 4, 4 0, 5 0, 5 4, 4 4, 4 0, 5 1, 1 5, 0 5, 0 1, 1 Domination in games without PRI s1 s2 t1 t2

  6. Equilibrium in games without PRI:PI Equilibrium 2, 8 5, 1 0, 5 3, 6 1, 5 6, 4 7, 2 1, 4 0, 2 5, 2 5, 0 2, 4 1, 1 6, 0 4, 2 3, 3

  7. PI versus Bayesian Equilibrium A strategy of i is a function bi that assigns an action xi to every possible type ti of i; bi(ti)=xi. A profile of strategies (b1,...,bn) is a PI equilibrium in a game without PRI if for every i, for every type of i, ti, and for every profile of types for the other agents, t-i, ui(ti,t-i,xi,b-i(t-i)) is maximized at the action xi=bi(ti). It is a Bayesian equilibrium w.r.t. a joint probability distribution P over the set of vector of types, (ti,t-i) if for every ti EP(ui(ti,t-i,xi,b-i(t-i))|ti) is maximized at the action xi=bi(ti). Up to some technicalities: b is a PI equilibrium iff it is a Bayesian equilibrium for every distribution of types.

  8. s1 s2 PI domination, and PI equilibrium with mixed strategies are also defined: 0.5 0.5 0.5 0.5 0.5 t1 0.5 0.75 t2 0.25

  9. Combinatorial Auctions: The (GVA) (VCG) (G) (C) Mechanism GVA= Generalized Vickrey Auction VCG= Vickrey-Clarke-Groves G=Groves C=Clarke The GVA game is described , in most cases, as a game without PRI The most used solution concept is Probabilistic Independent Domination A type of i is a valuation function ti defined on the set of all subsets of the set of goods A: ti(B) is the amount that i is willing to pay for the subset of goods B. bi(ti)=ti is a PI dominating strategy for i

  10. Moving to TEX

  11. Moving to Board

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