Attainability in Repeated Games with Vector Payoffs Eilon Solan Tel Aviv University Joint with: Dario Bauso, University of Palermo Ehud Lehrer, Tel Aviv University

Two players play a repeated game with vector payoffs which are d-dimensional. The total payoff up to stage n is G n. 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, G n /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, G n, converges to A, regardless of the strategy of player 2.

Motivation 1: Control theory d n is the demand at stage n (multi-dimensional, unknown). s n is the supply at stage n (multi-dimensional, controlled by the decision maker). s n – d n is 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. c n = bank's capital at stage n a n = bank’s risk-weighted assets at stage n. c n / a n = capital adequacy ratio at stage n. 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, G n, converges to A, regardless of the strategy of player 2.

A repeated game with vector payoffs that are d-dimensional (A 1, A 2, u). The Model The game is in continuous time. We consider non-anticipating behavior strategies with σ i = (σ i (t)) is a process with values in ∆(A i ), such that there is an increasing sequence of stopping times τ i1 < τ i2 < τ i3 < … that satisfies: For each t, τ ik ≤ t < τ i,k+1 σ i (t)) is measurable w.r.t. the information at time τ ik.

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

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 σ 1 that ensures that ∫ s=0 t g s (mixed action pair at time s)dslim t→∞ = x for every strategy σ 2 of player 2. The strategy σ 1, 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.

Theorem: the vector x is attainable by player 1 if and only if a) The vector 0 is attainable by player 1. b) For every function f : ∆(A 1 ) → ∆(A 2 ) the vector x is in the cone generated by { u(p,f(p)) : p in ∆(A 1 ) }. 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 ∆(A 1 ) and player 2 chooses f : ∆(A 1 ) → ∆(A 2 ). 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.

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

1) Characterization of attainable sets. Further Questions 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.