- Reinforcement learning.  Recurring chicken game
- The chicken game -
https://en.wikipedia.org/wiki/Chicken_(game)
- Nash equilibrium f_1(x, y) >= f_1(z, y), f_2(x, y) >= f_2(x, z)
- (2, 7), (7, 2) - nash equilibrium in the chicken game
- The minmax/maxmin is 2.  The player can ensure he get get something regardless of what the other is doing
-- The two player if she plays dare the worst that can happen is 0, if she plays chicken that worst that can happen is 2.   Then by playing chicken she can ensure 2.   Same with the column player
-  We'll repeat the game infinite number of times.   The payment of the row play is ri at stage.   Will define the payment to be the liminf of the average
Sum_{1 to k}(ri)/k as k goes to infinite.   Same thing with the column player

- The folk theorem in game theory - https://en.wikipedia.org/wiki/Folk_theorem_(game_theory)
- Objective - obtain an equilibrium of (1/2)(0,0) + (1/2)(6, 6).   In face any convex combination of the possible game payments can be obtained. 
-  Strategy players play (0 0) in odd stages and (6 6) in even stages.   If the column player changes from the prescribed behavior the row player play dare always ensuring the player gets no more than 2.   Same with the column player.  
-- Analysis
- Payments when nobody deviates - same for both players.   For row player the payment is

0 6 0 6 0 6 ....
0 3 2 3  12/5 3 ....

Liminf of the series is 3
https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

Assume a deviation of the column play then at the deviation the column player gets a maximal addition of 2 then moving forward after the deviation the column player is getting a maximum of 2

Payments of column player is bounded by -

0 6 0 6 2 2 2 2 2 2 ....

Assume the length of the prefix is 2k before the deviation then the payment for the column player is bounded by

Lim [k6 + (n - 2k)2]/n = lim 2 = 2

Which is less than 3. 

Therefore, the above strategy is a nash equilibrium . 

Can we marry repeated games with re-enforcement learning?  See additional interesting reading - 
https://arxiv.org/pdf/1703.02702.pdf