Ties
Summary and Introduction
1 PRO-RATA concave game
1.1 strict pure symmetrical balance
1.2 Unique balance
1.3 Gayof of creibriui
2 decentralized batch exchanges
2.1 Arbitration
3 Conclusion and references
A digital
B Additional numbers
C strict relaxing concavity
D ROSEN condition
1.1 strict pure symmetrical balance
There is a strict pure balance where all players have equal strategies, given by x = (q / n) 1 where Q is the optimizer of the following problem:
With the variable Q ∈ R. We will first show certain properties of this result, then show that the pure strategy x = (q / n) 1 is, indeed, a balance.
Discussion. It may seem that the condition placed on F is very strong, but in fact, any F not satisfying the above state has only trivial balances (or not). In particular, since F is concave, if f does not satisfy the above condition, either (a) f is strictly positive everywhere, except at f (0) = 0, (b) f is strictly negative everywhere, except at f (0) = 0, or (c) f = 0. In the first case, there is no balance because any player can improve their payment by increasing their strategy. In the second case, any player who plays a non -zero strategy receives a negative gain (while playing the zero strategy would give 0 gain). Although, in the third case, any strategy is a balance.
Balance properties. The collection of strategies x = (q / n) 1 is clearly pure and symmetrical. To see that x = (q / n) 1 is a strict balance, note that the best answer for any player I, when each other player plays the Q / N strategy is:
Then note that Q> 0 must fulfill the first -rate optimality conditions of (4):
1.2 Unique balance
Positivity of balances. We will first show that f (v)> 0 for each 0
1.3 Gayof of creibriui
Conditioned on each player receiving the same gain (a condition of equity), the optimal allowance that each player would obtain is
which is, by definition, at least as good as the equilibrium gain:
where Q> 0 is the solution to (4). In fact, we can show that the optimal allocation of the fair is always strictly better than equilibrium gain. To see this, note that, under the hypotheses on F introduced above, we know that sup f is obtained by a certain value 0
For all n> 1 since f (q)> 0. This means that q does not satisfy the condition of optimality to maximize f, therefore f (q)
Prices of anarchy. Given the same hypotheses as the beginning of §1.2 on the F function, it is not difficult to show that the price of the satisfied anarchy
As the number of players N becomes large for a C. constant to see it, consider the optimality conditions of first order for Y (4):
Note that F 0 (Q) <0 depuis q> 0 and f (q)> 0, so
Whenever n> 1. Since f is concave, then f 0 is monotonous non -increasing and, since q ≤ w for each n we have that
Finally, we know that sup f is constant in the number of players, so
