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 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.
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 – dn 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.
cn = bank's capital at stage n
an = bank’s risk-weighted assets at stage n.
cn / an = capital adequacy ratio at stage n.
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.
The Model
gt = payoff at time t (given the mixed actions of the players).
Gt =
∫
t
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 ε
the set B(A, ε) is strongly attainable by player 1.
B(A, ε) := { x : d(x,A) ≤ ε }
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
lim
t→∞
∫
t
gs (mixed action pair at time s)ds = x
s=0
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.
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 : ∆(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.
Theorem: the following conditions are equivalent:
a) The vector 0 is attainable by player 1.
b) One has vλ ≥ 0 for every λ in Rd, where vλ is the value of the
game projected in the direction λ.
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.
If (b) holds:
Player 1 plays in small intervals. In each interval he pushes the
payoff towards 0.
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.