Non-Strictly Determined Zero-Sum Games and Mixed Strategies
⇓ Selected Mixed Strategies (p1 = prob. of s1) for Player 1 (Row Player)
Basic Pure Strategy Matrix Selected Mixed Strategies (p2 = prob. of c1) for Player 2 (Col. Player)
|
|
c1 |
c2 |
p2 = 0 |
p2 =.2 |
p2 =.4 |
p2 =.5 |
p2 =.6 |
p2 =.8 |
p2 = 1 |
min |
|
s1 |
3 |
2 |
2.00 |
2.20 |
2.40 |
2.50 |
2.60 |
2.80 |
3.00 |
2.00 |
|
s2 |
1 |
4 |
4.00 |
3.40 |
2.80 |
2.50 |
2.20 |
1.60 |
1.00 |
1.00 |
|
p1 = 0 |
1.00 |
4.00 |
4.00 |
3.40 |
2.80 |
2.50 |
2.20 |
1.60 |
1.00 |
1.00 |
|
p1 = .2 |
1.40 |
3.60 |
3.60 |
3.16 |
2.72 |
2.50 |
2.28 |
1.84 |
1.40 |
1.40 |
|
p1 = .4 |
1.80 |
3.20 |
3.20 |
2.92 |
2.64 |
2.50 |
2.26 |
2.08 |
1.80 |
1.80 |
|
p1 = .6 |
2.20 |
2.80 |
2.80 |
2.68 |
2.56 |
2.50 |
2.44 |
2.32 |
2.20 |
2.20 |
|
p1 = .7 |
2.40 |
2.60 |
2.60 |
2.56 |
2.52 |
2.50 |
2.48 |
2.44 |
2.40 |
2.40 |
|
p1 = .75 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
2.50 |
|
p1 = .8 |
2.60 |
2.40 |
2.40 |
2.44 |
2.48 |
2.50 |
2.52 |
2.56 |
2.60 |
2.40 |
|
p1 = 1 |
3.00 |
2.00 |
2.00 |
2.20 |
2.40 |
2.50 |
2.60 |
2.80 |
3.00 |
2.00 |
|
max |
3.00 |
4.00 |
4.00 |
3.40 |
2.80 |
2.50 |
2.60 |
2.80 |
3.00 |
|
Darkest shading: Labels for rows and columns
Medium shading: Basic 2 × 2 payoff matrix (for pure strategies) [Note that this game is not strictly determined because maximin (for P1) = 2.00 < minimax (for P2) = 3.00, and (equivalently) because there is no equilibrium outcome.]
Lightest shading: Expected payoff for pure strategies vs. mixed strategies
No shading: expected payoff for mixed strategies vs. mixed strategies
A mixed strategy with p = 0 or p = 1 is called a “degenerate” mixed strategy.
Here is the expected payoff calculation for the mixed strategy pair (p1 = .2, p2 = .4)
[.2 × .4 × 3] + [.2 × .6 × 2] + [.8 × .4 × 1] + [.8 × .6 × 4]
[.08 × 3] + [.12 × 2] + [.32 × 1] + [.48 × 4]
.24 + .24 + .32 + 1.92 = 2.72
Chart version of Mixed Strategy Payoffs
In Chart I, the upward sloping solid line is the payoff for P1 (Row) for all of P1's mixed strategies running from [p = 0] to [p = 1] when P2 (Column) chooses pure strategy c1; likewise the downward sloping solid line is the payoff for P1 when P2 chooses pure strategy c2.
If P2 chooses a mixed strategy, P1's payoffs fall on a straight line “between” the two solid lines, e.g., on one of the dotted lines. It can be seen that P1's maximin mixed strategy is [p = .75], and that this mixed strategy guarantees P1 a higher payoff (2.50) than P1's maximin pure strategy does (2.00).
Chart II, interpreted in the same manner but from P2's point of view, shows that P2's minimax mixed strategy is [p = .5], and that this mixed strategy holds P1 down to P1's maximin payoff of 2.50., while P2's maximin pure strategy can hold P1's payoff down only to 3.00. Also, you can refer back to the payoff matrix to verify that, if P1 chooses his maximim strategy [p = .75] and P2 chooses his minimax strategy [p = .5], the outcome is an equilibrium.
A non-zero sum game such as this is said to be non-strictly determined, in that maximin for P1 = 2.50 = minimax for P2 and the resulting outcome is an equilibrium. It is only “non-strictly” determined because the equilibrium outcome is in mixed strategies rather than pure strategies.
The Minimax Theorem, often regarded as the foundation of game theory, demonstrates that all two-player zero-sum games are (at least) non-strictly determined.
