Mixed Strategies: Graphical

Jennifer Firkins Nordstrom · Introduction to Game Theory, Section 3.2 · CC BY-SA 4.0

Plot expected payoff against mixing probability p. Each of the opponent's pure strategies gives a line. The optimal mix is where the lines cross: the point where the opponent can't exploit your pattern.

The setup: a 2x2 zero-sum game

Take a game with no pure-strategy equilibrium. Player 1 mixes between rows A and B with probability p on B. For each of Player 2's pure strategies (column C or D), Player 1's expected payoff is a linear function of p. Plot both lines.

Scheme

; Payoff matrix (Player 1's payoffs, zero-sum):
;        C    D
;   A  [ 1    0 ]
;   B  [-1    2 ]
;
; Player 1 plays B with probability p, A with (1-p).
; Expected payoff vs Player 2's pure strategies:
;   E(vs C) = 1*(1-p) + (-1)*p = 1 - 2p
;   E(vs D) = 0*(1-p) + 2*p   = 2p

(define (ev-vs-C p) (+ (* 1 (- 1 p)) (* -1 p)))
(define (ev-vs-D p) (+ (* 0 (- 1 p)) (* 2 p)))

; Sample points
(display "p=0.0: vs C = ") (display (ev-vs-C 0.0))
(display ", vs D = ") (display (ev-vs-D 0.0)) (newline)

(display "p=0.5: vs C = ") (display (ev-vs-C 0.5))
(display ", vs D = ") (display (ev-vs-D 0.5)) (newline)

(display "p=1.0: vs C = ") (display (ev-vs-C 1.0))
(display ", vs D = ") (display (ev-vs-D 1.0)) (newline)

Finding the intersection

Set the two expected-payoff lines equal: 1 - 2p = 2p. Solve: p = 1/4. Player 1 should play B with probability 1/4, A with probability 3/4. The expected payoff is 1/2 regardless of what Player 2 does. That's the guarantee.

Scheme

; Find the intersection: solve E(vs C) = E(vs D)
; 1 - 2p = 2p => 1 = 4p => p = 1/4

(define (find-optimal-p ev1 ev2)
  ; Binary search for intersection
  (let loop ((lo 0.0) (hi 1.0) (n 50))
    (if (= n 0) (/ (+ lo hi) 2.0)
        (let ((mid (/ (+ lo hi) 2.0)))
          (if (> (- (ev1 mid) (ev2 mid)) 0)
              (loop mid hi (- n 1))
              (loop lo mid (- n 1)))))))

(define p* (find-optimal-p ev-vs-C ev-vs-D))
(display "Optimal p = ") (display p*) (newline)
(display "Expected payoff = ") (display (ev-vs-C p*)) (newline)
(display "Check vs D:       ") (display (ev-vs-D p*))
; Both lines give 0.5 at p = 0.25

Why the intersection is optimal

Below the intersection, the minimum payoff comes from one line. Above it, from the other. The maximum of the minimum is at the crossing point. This is the maximin in action: Player 1 chooses the mix that makes the worst case as good as possible.

Scheme

; The maximin principle: maximize your minimum payoff
; At any p, the opponent picks whichever column hurts you more.
; Your guaranteed payoff = min of the two lines.

(define (guaranteed-payoff p)
  (min (ev-vs-C p) (ev-vs-D p)))

; Scan across p to find the best guarantee
(display "p=0.00: guarantee = ") (display (guaranteed-payoff 0.0)) (newline)
(display "p=0.10: guarantee = ") (display (guaranteed-payoff 0.1)) (newline)
(display "p=0.25: guarantee = ") (display (guaranteed-payoff 0.25)) (newline)
(display "p=0.50: guarantee = ") (display (guaranteed-payoff 0.5)) (newline)
(display "p=1.00: guarantee = ") (display (guaranteed-payoff 1.0)) (newline)
; Best guarantee is at p=0.25, payoff=0.5

Notation reference

Textbook	Scheme	Meaning
p	p	Probability Player 1 plays row B
E(vs C)	(ev-vs-C p)	Expected payoff when Player 2 plays C
E(vs D)	(ev-vs-D p)	Expected payoff when Player 2 plays D
maximin	(guaranteed-payoff p)	Minimum payoff across opponent's choices

Neighbors

Nordstrom sequence

🎲 Nordstrom 3.1 — Repeated Games (previous)
🎲 Nordstrom 3.3 — Mixed Strategies: Expected Value (next)

🎲 Nordstrom 2.5 — Equilibrium Points (pure strategy version)
🍞 Hedges 2018 — Compositional Game Theory (where mixed strategies live categorically)
🧠 Lovelace Ch.5 Reinforcement Learning — exploration-exploitation tradeoffs use mixed strategies

Foundations (Wikipedia)

Translation notes

The graphical method only works for 2x2 games because we need exactly two lines in a 2D plot. For larger games, use the expected value method (next page) or linear programming. For a geometric approach to strategy spaces in auctions, see power diagrams applied to ad auctions. The payoff matrix here uses Nordstrom's example from Section 3.2.

Want the full treatment? Read Section 3.2 in Nordstrom's textbook.

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