Noncooperative Games, Couplings Constraints, and Partial Effi ciency

Transcription

1 Noncooperative Games, Couplings Constraints, and Partial Effi ciency Sjur Didrik Flåm University of Bergen, Norway

2 Background Customary Nash equilibrium has no coupling constraints. Here: coupling constraints are crucial; sharing of input/output. Exchange is part of the game; driven by gradient differences. Walras equilibrium in exchange markets is part of the solution. Agent-based adjustments; no coordination. Myopic and adaptive behavior. Still: convergence to equilibrium.

3 Issues Typically, Nash equilibrium isn t Pareto effi cient. But, Walras equilibrium is! (the so-called welfare theorems). Hence only partial effi ciency. Where do market prices come from? Is price-taking behavior meaningful? Personal price willingness to pay. Is full optimization & perfect foresight needed?

4 The Game Player i I uses strategy x i X i (a Euclidean space) gets payoff π i (x i, x i ) R { } is subject to coupling constraints: (x i ) =: x X X := Π i I X i where X Π i I P i X. Nash equilibrium x π i (x i, x i ) = max {π i (ˆx i, x i ) : (ˆx i, x i ) X } > i. Normal (Nash) equilibrium x { } x arg max π i (ˆx i, x i ) : ˆx X i I >.

5 On normal equilibrium Existence of normal equilibrium is guaranteed when X is non-empty compact convex, and π(ˆx, x) := π i (ˆx i, x i ) i I is concave in ˆx, usc in (ˆx, x), & continuous in x. Characterization of normal equilibrium x comes in 3 equivalent ways: * There is a margin m M(x) N(x) where M(x) := ˆx π(ˆx, x) ˆx =x and N(x) := N(x, X ) := normal cone to X at * Each feasible deviation d D(x) := R + (X x) clr + (X x) =: T (x) = tangent cone satisfies the variational inequality m, d 0. * The orthogonal projection of m onto the tangent cone is nil: 0 = P T (x ) [m]

6 Out-of-equilibrium adjustments Is the differential inclusion ẋ P T (x ) [M(x)], x(0) X, (1) a reasonable model of disequilibrium play? One appeal: It s driven by local, decentralized data M i (x) := ˆx i π i (ˆx i, x i ) ˆxi =x i Two drawbacks: First, continuous time isn t quite realistic. Second, it s not decentralized! However, (1) helps in identifying a stability condition: a normal equilibrium x is asymptotically stable - hence unique - under (1) if (2) holds when x X x implies x X x = M(x), x x < 0. (2) π( x, x) > π(x, x) or M(x) M( x), x x < 0.

7 Coupling constraints Henceforth X = { x = (x i ) : x i = (y i, z i ) Y i Z i & i I } z i e i. i I Y i, Z i are non-empty compact convex Euclidean spaces Y i, Z. Agent i owns endowment e i. π i (x i, x i ) = π i (y i, y i, z i ). Increasing in z i for at least one i. A workhorse instance: Cournot oligopoly (1838): Firm i I produces y i, uses production factors z i, incurs cost c i (y i, z i ), and gets payoff π i (y i, y i, z i ) = P(y I ) y i c i (y i, z i ) where y I := y i. i I

8 The solution concept A feasible profile x = (x i ) X and a price vector p Z is a Nash-Walras equilibrium iff x i = (y i, z i ) maximizes π(ŷ i, y i, ẑ i ) p, ẑ i s. t. (ŷ i, ẑ i ) Y i Z i i, and all exchange markets clear: p 0, i I z i e i, i I and p, i I e i z i = 0. i I

9 Getting around projection Recall: in ẋ P T (x ) [M(x)], x(0) X, projection does not square with decentralized play. Idea: Since coupling constraints are fairly simple - in fact, additive - bilateral exchange might take care of projection.

10 Where does the price come from? Recall that in maximizing π i (ŷ i, y i, ẑ i ) p, ẑ i s. t. (ŷ i, ẑ i ) Y i Z i agent i must predict y i and p. In particular, where did p come from? How could p be learned? Idea: Again, bilateral exchanges become instrumental. These are likely to generate p.

11 Repeated play under additive coupling There are two time scales (or clocks). The fast one secures that That is, at any time π i (y i, y i, z i ) = max π i (Y i, y i, z i ) i y Nash(z). The slow one regulates updates of z = (z i )

12 Repeated play unfolding as a discrete-time algorithm Start at some feasible allocation i z i Z i of transferable goods. Find contingent Nash equilibrium y Nash(z). Two agents i, j meet with uniform probability, holding z i Z i and z j Z j respectively. Update their holdings so that z i +1 := z i + sd ij Z i and z j +1 := z j sd ij Z j with step size s 0 and direction [ ] d ij π i (y, z i ) N(z i, Z i ) z i [ ] π j (y, z j ) N(z j, Z j ). z j Continue to find contingent Nash equilibrium until convergence.

13 Convergence Theorem Suppose repeated play, as modelled above, proceeds at discrete stages k = 0, 1,... with step sizes s k 0 that satisfy s k = + k=0 and sk 2 < +. k=0 Then, under asymptotic stability, the generated sequence x k, k = 0, 1,... converges to the unique Nash-Walras equilibrium.

14 Upshot so far There are several agents, each holding his (stochastic) endowment = (random) resource bundle = contingent claim. They proceed (along Nash equilibrium) by repeated bilateral barters. Issues: Will full Nash-Walras equilibrium obtain? Is there room for "simple" agents? Quit likely: Yes! Is full optimization & perfect foresight really needed? Must prices be announced? Yes! No! No!

15 In line with behavioral economics Agents lack computational competence, perfect foresight, information, global vision,... There is no coordination, no auctioneer, no market maker,... Bargaining, matching, search is not made explicit - and not really needed. Nonetheless: strategies & holdings converge to equilibrium. Inspiration from Pareto:"The economy is a great computing machine."

16 Back to bilateral barters Agent i meets agent j. Their actual "states" are x i = (y i, z i ) and x j = (y j, z j ). Fix y and posit Π i (z i ) := π i (y, z i ), Π j (z j ) := π j (y, z j ). Recall that z i +1 := z i + sd ij Z i and z j +1 := z j sd ij Z j with step size s 0 and direction [ ] d ij Π i (z i ) N(z i, Z i ) z i What is the nature of (a good direction) d ij? [ ] Π j (z j ) N(z j, Z j ). z j

17 Intermezzo: the nature of the Walrasian part of equilibrium Definition A feasible allocation (z i ) of e I := i I e i and a linear price p : Z R constitute a Walras equilibrium iff Π i (z i ) + p(e i z i ) Π i (ẑ i ) + p(e i ẑ i ) for each i I and ẑ i Z i. In equilibrium each agent maximizes his payoff + his net value of sales. Proposition (On Walras equilibrium) A profile (z i ) alongside a price p constitutes a Walras equilibrium iff p Π i (z i ) N i (z i ) for each i, and z i = e I. i I that is: there is a common price price "=" marginal utility (modulo normal components)

18 More on the nature of equilibrium: Cooperative aspects Suppose coalition C I uses aggregate endowment e C := i C e i to get (sup-convolution) Π C (e C ) := sup { i C Π i (z i ) : i C z i = e C & z i Z i }. Payment scheme (π i ) is in the core of this transferable-utility game iff { Pareto effi cient: i I π i = Π I (e I ) stable against blocking: i C π i Π C (e C ) for each C I. Proposition (Equilibrium as core solution) For any equilibrium price p the payment scheme i π i := sup {Π i (ẑ i ) + p(e i ẑ i ) : ẑ i Z i } is in the core.

19 Maintaining feasibility throughout Cone of feasible directions for agent i at z i : Cone of common directions D i (z i ) := R + (Z i z i ). D ij (z i, z j ) := D i (z i ) D j (z j ). Tangent cone T ij (z i, z j ) := cld ij (z i, z j ) Maximal slope of joint improvement { S ij (z i, z j ) := max Π i (z i ; d) + Π j (z j ; d) : d T ij (z i, z j ) & d 1 }. d

20 On the maximal slope { S ij (z i, z j ) := max Π i (z i ; d) + Π j (z j ; d) : d T ij (z i, z j ) & d 1 }. d Proposition Let P ij = orthogonal projection onto T ij (z i, z j ) : S ij (z i, z j ) = min { P ij [g i g j ] : g i Π i (z i ), g j Π j (z j )} = dist [ Π i (z i ) N i (z i ), Π j (z j ) N j (z j )], where N i (z i ) is the normal cone to Z i at z i.

21 Material transfers Fix an Armijo parameter ϕ ij (0, 1) for each agent pair i, j. Definition: While holding z i Z i and z j Z j, agents i, j, make a real transfer sd, with s > 0 and d = 1, if and satisfies z i + sd Z i and z j sd Z j Π ij := Π i (z i + sd) + Π j (z j sd) Π i (z i ) Π j (z j ) Π ij s ϕ ij S ij (z i, z j ). Proposition When S ij (z i, z j ) > 0, there exists a real transfer.

22 Monetary transfers = side payments Π ij > 0 = side payments r i and r j such that r i + r j = 0 and is solvable. Π i (z +1 i ) + r i > Π i (z i ) & Π j (z +1 j ) + r j > Π j (z j ) Money oils the transaction machinery. Deals and incentives are compatible.

23 On common price and joint improvement Recall that in equilibrium p i I [ Π i (z i ) N i (z i )]. We say agents i, j see a common price iff [ Π i (z i ) N i (z i )] [ Π j (z j ) N j (z j )] =. Proposition i, j see a common price iff S ij (z i, z j ) = 0. Thus improvement is possible as long as some S ij (z i, z j ) > 0.

24 Asymmetric information Z = L 0 (S, F, µ, E) with F finite. Z i = L 0 (S, F i, π, E) with F i F. and d T ij (z i, z j ) = Z i Z j = z i Z i = D i (z i ) = Z i. d constant on A i A j when atoms A i F i and A j F j intersect. Projection = conditional expectation: Pr(z) s = s A z s µ s s A µ s for each s atom A.

25 Concluding remarks on equilibrium and dynamics? How can players arrive at equilibrium - if any? While underway, how much competence, coordination, and foresight is required? What are the roles of cognition and perception?

26 Play as viewed here It requires no coordination, experience, foresight, or optimization,.. It s totally decentralized. It s fully driven by low-complexity adaptive agents.

27 References Bauschke & Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review (1996) Feldman, Bilateral trading processes, pairwise optimality, and Pareto optimality, Rev Econ Studies (1973) Flåm &Koutsougeras, Private Information, Transferable Utility, and the Core, Economic Theory (2010) Flåm, On sharing of risks and resources, in Reich & Zaslavksi Optim Th. and Related Topics, Am Math Soc, Contemp Math (2012) Flåm, Exchanges and measures of risk, Math and Financial Econ (2012) Flåm et al, Gradient Differences and Bilateral Barters, Optimization (2012) Flåm, Coupled Projects, Core Imputations, and the CAPM, J. Math Econ (2012) Flåm & Gramstad, Direct exchanges in linear economies, Int. J. Game Th (2013) Gode & Sunder, Allocative effi ciency of markets with zero intelligence traders: markets as a partial substitute for individual rationality, J. Pol. Econ (1993) Shapley & Shubik, Trade using one commodity as a means of payment, J Pol Econ (1977)