Content map: SMU H3 Game Theory Map

Chapter 7: Mixed Strategies

Motivation for Mixed Strategies

Definition

Definition:

A mixed-strategy equilibrium is a Nash equilibrium in which at least one player uses a probability distribution over pure strategies. The definition of Nash equilibrium itself is unchanged.

Method / Reasoning

Write the situation as a simultaneous-move game.
Let each player randomise across pure actions with explicit probabilities.
Compute expected payoffs from the pure actions in the support.
Solve for probabilities that make players willing to mix.

Implications

Pure-strategy analysis may miss stable outcomes.
Strategic unpredictability can be an equilibrium object rather than a behavioural mistake.

Result:

Mixed-strategy analysis extends Nash equilibrium from deterministic play to probability distributions over actions.

Insight:

Mixing is deliberate randomisation, not indecision.

Probability Rules and Expected Value

Concept

Mixed strategies are evaluated through probabilities over possible outcomes.
Probability can be seen as long-run frequency over many observations.

Formal Definition

Definition:

If the possible outcomes are $x_1,\dots,x_n$ with probabilities $p_1,\dots,p_n$ , then

0 \leq p_i \leq 1, \qquad \sum_{i=1}^{n} p_i = 1.

Product Rule

Multiply probabilities when the target event is a specific sequence of independent draws.
Example:

P(HHT)=P(H)\cdot P(H)\cdot P(T)=\tfrac{1}{2}\cdot\tfrac{1}{2}\cdot\tfrac{1}{2}=\tfrac{1}{8}.

In mixed strategies, this gives the probability of one exact action profile.

Summation Rule

Add probabilities when several disjoint cases generate the same target event.
Example: rolling a total of $5$ with two dice:

P(5)=P(1,4)+P(2,3)+P(3,2)+P(4,1)=\tfrac{4}{36}.

In mixed strategies, this combines distinct opponent action profiles that lead to the same payoff case.

Expected Value

Definition:

For numerical outcomes, expected value is

E[X] = \sum_{i=1}^{n} p_i x_i.

Expected value is the probability-weighted average of the possible payoffs.
Compute it only after the relevant case probabilities are correct.
In mixed-strategy games, compare actions using expected payoff, not one realised draw.

Implications

Expected payoff, not realised payoff in one draw, is the relevant comparison for mixed strategies.
Case-by-case probability calculations feed directly into equilibrium conditions.

Result:

Expected payoff is the payoff concept used to compare mixed strategies.

Insight:

Correct event probabilities turn verbal randomisation into solvable equilibrium equations.

Pure and Mixed Strategies

Concept

A player may either choose one action for sure or randomise across several actions.
The notes describe randomisation as commitment to a random device.

Formal Definition

Definition:

A pure strategy chooses one action with probability $1$ .

A mixed strategy is a probability distribution over pure strategies.

For a two-action game, a typical mixed strategy is $(p,1-p)$ with $p\in(0,1)$ .

Implications

Every pure strategy is a special case of a mixed strategy.
Randomisation is useful when a predictable pure action can be exploited.

Insight:

What matters is not randomness by itself, but whether randomness removes the opponent’s gain from prediction.

Indifference Principle and Support Conditions

Concept

Players mix because they want to stay unpredictable.
The equilibrium test is still Nash equilibrium, but the verification now uses expected payoffs.

Formal Definition

Definition:

There are two conditions for mixed-strategy equilibria:

PR1: If a player mixes across several pure strategies, all pure strategies in the mixing bunch must yield the same expected payoff.

PR2: Any pure strategy outside the mixing bunch cannot yield a strictly higher expected payoff.

Method / Reasoning

Guess the support.
Write the expected payoff from each pure strategy in that support.
Impose equality across supported actions.
Check that excluded actions do not yield more.

Implications

The opponent’s mixing probabilities are pinned down by your indifference conditions.
Solving equalities is only the first step; support verification is essential.

Result:

Indifference conditions determine equilibrium mixing probabilities.

Insight:

A player mixes only when the supported pure strategies tie in expected payoff.

Computing Mixed Equilibria

Concept

Solve mixed equilibria by conditioning on exhaustive events.
Symmetric mixing is handled by letting each player use the same probability $p$ for one focal action.

Formal Definition

Definition:

For any pure strategy $s_i$ , expected payoff is

E[\pi_i(s_i)] = \sum_{k=1}^{n} p_k \cdot \pi_{i, k}

where $s_k$ are the opponents’ strategies with associated probability $p_k$

Method / Reasoning

Write the payoff matrix for the game.
Assume player 1 mixes with probabilities $(p, 1-p)$ and player 2 mixes with probabilities $(q, 1-q)$ .
Compute expected payoff for each pure action against the opponent’s mix.
Apply indifference: set expected payoffs equal for actions in the support.
Solve the resulting system for $p$ and $q$ .
Verify support conditions (PR2) for any excluded actions.

Example

	L ( $q$ )	R ( $1-q$ )
U ( $p$ )	3, 1	0, 2
D ( $1-p$ )	1, 3	2, 0

Suppose player 1 mixes $(p, 1-p)$ on $(U,D)$ and player 2 mixes $(q, 1-q)$ on $(L,R)$ .
Expected payoff for player 1 from $U$ :

E[\pi_1(U)] = 3q + 0(1-q) = 3q

Expected payoff for player 1 from $D$ :

E[\pi_1(D)] = 1q + 2(1-q) = 2 - q

Indifference: $3q = 2 - q \Rightarrow q = \tfrac{1}{2}$ .
Expected payoff for player 2 from $L$ :

E[\pi_2(L)] = 1p + 3(1-p) = 3 - 2p

Expected payoff for player 2 from $R$ :

E[\pi_2(R)] = 2p + 0(1-p) = 2p

Indifference: $3 - 2p = 2p \Rightarrow p = \tfrac{3}{4}$ .

Result:

As such, the mixed equilibrium is:

\{\left(\tfrac{3}{4}, \tfrac{1}{4}\right), \left(\tfrac{1}{2}, \tfrac{1}{2}\right)\}

Best Response Correspondences

Concept

Best responses in mixed games are correspondences rather than single actions.
At some opponent probabilities, a player may be willing to use any mixture inside a support.

Formal Definition

Definition:

A best response correspondence maps the opponent’s mixed strategy into the set of all optimal pure or mixed responses.

Method / Reasoning

Express each pure strategy’s expected payoff as a function of the opponent’s mix.
Compare those payoff functions point by point.
If one pure action does better, choose it with probability $1$ .
If two supported pure actions tie, any mixture across them is a best response.

Implications

Equilibria remain intersections of best responses.
To find all equilibria, perform best-response analysis and then check support conditions.

Result:

Graphical best-response analysis remain the standard way to locate mixed equilibria.

Zero-Sum Games and Minimax Logic

Concept

In zero-sum games one player’s gain is the other player’s loss.
Predictable pure play is often exploitable, so mixing becomes central.

Method / Reasoning

Write the payoff matrix for one player.
Choose probabilities that make the opponent indifferent.
Solve for the value of the game.

Implications

Equilibrium predicts frequencies of play rather than one deterministic move.
The equilibrium value summarises the expected performance level under optimal play.

Result:

In zero-sum games, equilibrium mixing equalises the opponent’s supported payoffs and prevents exploitation.

Insight:

Minimax logic is defensive, it chooses probabilities that make you hard to exploit.

Game Theory Chapter 7: Mixed Strategies

Chapter 7: Mixed Strategies

Motivation for Mixed Strategies

Definition

Method / Reasoning

Implications

Probability Rules and Expected Value

Concept

Formal Definition

Product Rule

Summation Rule

Expected Value

Implications

Pure and Mixed Strategies

Concept

Formal Definition

Implications

Indifference Principle and Support Conditions

Concept

Formal Definition

Method / Reasoning

Implications

Computing Mixed Equilibria

Concept

Formal Definition

Method / Reasoning

Example

Best Response Correspondences

Concept

Formal Definition

Method / Reasoning

Implications

Zero-Sum Games and Minimax Logic

Concept

Method / Reasoning

Implications