Links table
Abstract
1 Pro-Rata Game
1.1 Miserable, pure balance
1.2 The balance is unique
1.3 balance returns
2 double decentralized exchanges
2.1 The pleading
3 conclusion and references
Numbers
In additional numbers
C relaxation
Rosen case
1.1 Miserable, pure balance
There is a strict and pure balance where all players have equal strategies, provided by x = (Q/N) 1 where the Q is improved the following problem:
With the Q ∈ R variable, we will show some characteristics of this result first, then we show that the pure strategy = (Q/N) 1 is actually a balance.
discussion. It may seem that the condition placed on F is very strong, but in reality, any F does not satisfy the above condition has a trivial balance (or not) only. In particular, given that F is concave, if F does not satisfy the aforementioned condition, (A) F are completely positive everywhere except F (0) = 0, (B) F completely negative everywhere except for F (0) = 0, or (C) F = 0. In the second case, any player who plays a non -zero strategy receives a negative reward (while playing the zero strategy will give a reward 0). Whereas, in the third case, any strategy is a balance.
Balance properties. Collect the x = (Q/N) 1 pure and clearly similar. To know that x = (Q/N) 1 is a strict balance, note that the best response to any player I, when each other player plays the Q/N strategy:
Next, note that Q> 0 should fulfill the ideal conditions of the first degree (4):
1.2 The balance is unique
Positive balance. First, we will appear that F (V)> 0 per 0
1.3 balance returns
Conditioned for every player who receives the same reward (fairness), the optimal allocation that each player will receive is
It is, by definition, at least good as the balance of balance:
Where q> 0 is the solution for (4). In fact, we can show that optimal fair allocation is always completely better than balance. To find out, note that according to the assumptions in F above the above, we know that SUP F is realized by some value 0
For everyone n> 1 because F (Q)> 0. This means that Q does not fulfill the optimal state of the F., so F (Q)
Chaos price. Looking at the same assumptions as the beginning of paragraph 1.2 on the F -Fun
Since the number of players N becomes large for some C. fixed to see this, think about the terms of the optimal first degree of Y (4):
Note that F 0 (Q) <0 منذ q> 0 and f (Q)> 0, so
Whenever N> 1. Since F is concave, F 0 is not increasingly intense, and since Q ≤ w for each n, we have that.
Finally, we know that SUP F is fixed in the number of players, so
