Another "tennis" game. Let the payoff matrix be:
|
L |
R |
U |
80,30 |
20,50 |
D |
40,70 |
60,10 |
There's no pure NE, or NE with either of the players playing pure strategies (why?), so let $(1-p,p)$ indicate player 1 (Row)'s probabilities of playing (U,D),
and $(1-q,q)$ indicate player 2 (Column)'s probabilities of playing (L,R).
- Write down expressions, involving $q$, for Row's payoffs 1) when playing U and 2) when
playing D, equate them, and solve for $q$.
- Similarly, write down expressions for Column's pure payoffs and solve for $p$.
- Plot Row's payoffs under each pure strategy, as a function of $q$ (that is,
two straight lines, where the x axis is $q$, ranging from 0 to 1, and the y axis is $u_1$). Also, plot Row's BR as a
function of $q$ (x axis is $q$, y axis is the value $p$ of Row's BR).
- Similarly, plot Column's payoffs under its pure strategies, and plot its BR.
- Jointly plot the BRs, indicating their intersection (x axis $q$, y axis $p$).