Attainability in Repeated Games with Vector Payoffs

1. Attainability in Repeated Games with Vector Payoffs EilonSolan Tel Aviv University Joint with: Dario Bauso, University of Palermo Ehud Lehrer, Tel Aviv University

2. Two players play a repeated game with vector payoffs which are d-dimensional. The total payoff up to stage n is Gn. Definition (Blackwell, 1956): A set of payoff vectors A is approachable by player 1 if player 1 has a strategy such average payoff up to stage n, Gn/n, converges to A, regardless of the strategy of player 2. Definition: A set of payoff vectors A is attainable by player if player 1 has a strategy such that the total payoff up to stage n, Gn, converges to A, regardless of the strategy of player 2.

3. Definition: A set of payoff vectors A is attainable by player if player 1 has a strategy such that the total payoff up to stage n, Gn, converges to A, regardless of the strategy of player 2. Motivation 1: Control theory dn is the demand at stage n (multi-dimensional, unknown). sn is the supply at stage n (multi-dimensional, controlled by the decision maker). sn – dnis the excess supply, the amount that is left in our storeroom. We need to bound the total excess supply. Motivation 2: Banking, Capital Adequacy Ratio. cn = bank's capital at stage n an = bank’s risk-weighted assets at stage n. cn / an= capital adequacy ratio at stage n.

4. The Model A repeated game with vector payoffs that are d-dimensional (A1, A2, u). The game is in continuous time. We consider non-anticipating behavior strategies with σi = (σi(t)) is a process with values in ∆(Ai), such that there is an increasing sequence of stopping times τi1 < τi2 < τi3 < … that satisfies: For each t, τik ≤ t < τi,k+1 σi(t)) is measurable w.r.t. the information at time τik.

5. The Model gt = payoff at time t (given the mixed actions of the players). t ∫ Gt = gs (mixed action pair at time s)ds s=0 Definition: A set A in Rd is strongly attainable by player 1 if player 1 has a strategy that guarantees that the distance limt→∞ d(A,Gt) = 0, regardless of player 2’s strategy. Definition: A set A in Rd is attainable by player 1 if for every εthe set B(A, ε) is strongly attainable by player 1. B(A, ε) := { x : d(x,A) ≤ ε }

6. Theorem: the set of vectors attainable by player 1 is a closed and convex cone. If the vector x is attainable Then there is a strategy σ1 that ensures that t ∫ lim gs (mixed action pair at time s)ds = x s=0 t→∞ for every strategy σ2 of player 2. The strategy σ1, accelerated by a factor of β, attains x/β. If the vectors x and y are attainable, to attain x+y, first attain x, then forget past play and attain y.

7. Theorem: the vector x is attainable by player 1 if and only if • The vector 0 is attainable by player 1. • For every function f : ∆(A1) → ∆(A2) the vector x is in the cone generated by { u(p,f(p)) : p in ∆(A1) }. If (b) does not hold: Player 2 plays f(α) whenever player 1 plays the mixed aciton α. If (a) + (b) hold: Consider an auxiliary one shot-game game in which player 1 chooses a distribution over ∆(A1) and player 2 chooses f : ∆(A1) → ∆(A2). For every strategy of player 2, player 1 has a response such that the average payoff is x. Therefore player 1 has a strategy that “pushes towards x” whatever f player 2 chooses.

8. If (b) does not hold: There is q in ∆(A2) such that the payoff is in some open halfspace. If Player 2 always plays this q, the payoff does not converge to 0. • Theorem: the following conditions are equivalent: • The vector 0 is attainable by player 1. • One has vλ≥ 0 for every λ in Rd, where vλ is the value of the game projected in the direction λ. If (b) holds: Player 1 plays in small intervals. In each interval he pushes the payoff towards 0.

9. Further Questions 1) Characterization of attainable sets. 2) Characterization of strongly attainable sets and vectors. 3) Characterization of attainable sets in discrete time. 4) Characterization of attainable sets when payoff is discounted.