Mixed Strategies Nash Equilibrium: The Math

1. Audio
   Observation

   The speaker states that Player 1 goes right with probability p and left with probability 1-p.

2. Diagram
   Observation

   The whiteboard shows 'p' next to the top row for Player 1's strategy R.

p

Probability that Player 1 chooses strategy Right (R).

[0, 1]

1. Audio
   Observation

   The speaker states that Player 2 goes right with probability q and left with probability 1-q.

2. Diagram
   Observation

   The whiteboard shows 'q' above the left column for Player 2's strategy R.

q

Probability that Player 2 chooses strategy Right (R).

[0, 1]

1. Formula
   Observation

   The speaker writes the equation $\pi_2(R) = p(1) + (1-p)(0) = p$.

$\pi_2(R)$

Expected payoff to Player 2 when choosing strategy Right (R).

Real numbers

1. Formula
   Observation

   The speaker writes the equation $\pi_2(L) = p(0) + (1-p)(2) = 2 - 2p$.

$\pi_2(L)$

Expected payoff to Player 2 when choosing strategy Left (L).

Real numbers

1. Diagram
   Observation

   The payoff matrix labels Player 1's row strategies as R and L.

R

Player 1's right-row pure strategy in the 2x2 payoff matrix.

Player 1 strategy set

1. Diagram
   Observation

   The payoff matrix labels Player 1's row strategies as R and L.

L

Player 1's left-row pure strategy in the 2x2 payoff matrix.

Player 1 strategy set

1. Diagram
   Observation

   The payoff matrix labels Player 2's column strategies as R and L.

2. Audio
   Observation

   At 01:00-01:09 the speaker says Player 2 mixes between right and left with probability q on right.

R

Player 2's right-column pure strategy.

Player 2 strategy set

1. Diagram
   Observation

   The payoff matrix labels Player 2's column strategies as R and L.

L

Player 2's left-column pure strategy.

Player 2 strategy set

1. Diagram
   Observation

   The board heading reads 'Player 2's choice of q:' and the equations use q and (1-q).

2. Audio
   Observation

   At 00:04-00:08 the speaker says, 'Now let's figure out player two's choice of q.'

q

Probability that Player 2 chooses R; then (1-q) is the probability of choosing L.

$0 \le q \le 1$

1. Diagram
   Observation

   The board writes $\Pi_1(R)=q(0)+(1-q)3=3-3q$.

2. Audio
   Observation

   At 00:15-00:29 the speaker sets up Player 1's payoff from going right.

$\Pi_1(R)$

Expected payoff to Player 1 when Player 1 plays R against Player 2's mixed strategy.

Real-valued expected payoff

1. Diagram
   Observation

   The board writes $\Pi_1(L)=q(4)+(1-q)0=4q$.

2. Audio
   Observation

   At 00:33-00:45 the speaker sets up Player 1's payoff from choosing left.

$\Pi_1(L)$

Expected payoff to Player 1 when Player 1 plays L against Player 2's mixed strategy.

Real-valued expected payoff

1. Diagram
   Observation

   The visible 2x2 matrix entries are (0,1), (3,0), (4,0), (0,2).

$\begin{bmatrix}0&3\\4&0\end{bmatrix}$,\ $\begin{bmatrix}1&0\\0&2\end{bmatrix}$

Row-player payoffs and column-player payoffs for the displayed game.

Payoff matrices for a 2x2 normal-form game

1. Audio
   Observation

   The speaker explains that players choose between going right or left with specific probabilities.

2. Diagram
   Observation

   The whiteboard displays a 2x2 payoff matrix with probabilities p, 1-p, q, and 1-q assigned to the rows and columns.

In a mixed strategy, a player randomizes over their available pure strategies. Each pure strategy is chosen with a specific probability, and the sum of these probabilities must equal 1.

1. Applies to games where players have multiple pure strategies.

2. Probabilities must be non-negative and sum to 1.

1. Audio
   Observation

   The speaker states the rule: 'Player 1 chooses p... in order to make Player 2 indifferent between going right and left.'

2. Diagram
   Observation

   The text on the whiteboard reads: 'Player 1's choice of p: Make Player 2 indifferent between R & L.'

To find a mixed strategy Nash equilibrium, a player selects the probabilities of their actions such that the opponent becomes indifferent between their own available pure strategies. This means the expected payoffs for the opponent's pure strategies must be equal.

1. Used when solving for mixed strategy Nash equilibria in finite games.

2. Requires calculating expected payoffs based on the opponent's mixed strategy.

1. Mixed Strategy Definition

1. Audio
   Observation

   At 00:08-00:15 the speaker says, 'For player two's choice of q, we need to make player one indifferent between choosing right and choosing left.'

2. Diagram
   Observation

   The board first writes two expected-payoff expressions and then sets them equal.

To find the probability with which Player 2 should mix so that Player 1 is willing to randomize, compute Player 1's expected payoff from each pure row strategy under Player 2's mixture, then equate those payoffs. The resulting q makes Player 1 indifferent between R and L.

1. The target player must be made indifferent between the pure strategies being mixed.

2. The opponent's probabilities must sum to 1.

3. The solution should lie in [0,1] to represent a valid mixed strategy.

1. Diagram
   Observation

   The board writes $\Pi_1(R)=q(0)+(1-q)3=3-3q$.

2. Audio
   Observation

   At 00:22-00:29 the speaker explains that if Player 1 goes right, Player 1 gets either 0 or 3 with the relevant probabilities.

When Player 1 chooses the top row R, the payoff is 0 if Player 2 chooses R and 3 if Player 2 chooses L. With Player 2 mixing using probability q on R, the expected payoff is the weighted average of those two outcomes.

1. Player 2 plays R with probability q and L with probability 1-q.

2. Payoffs are read from Player 1's row R in the displayed matrix.

1. Indifference method for finding an opponent's mixing probability

1. Diagram
   Observation

   The board writes $\Pi_1(L)=q(4)+(1-q)0=4q$.

2. Audio
   Observation

   At 00:37-00:45 the speaker explains that if Player 1 chooses left, Player 1 gets either 4 or 0 with the probabilities related to q.

When Player 1 chooses the bottom row L, the payoff is 4 if Player 2 chooses R and 0 if Player 2 chooses L. Under Player 2's mixture, the expected payoff is the weighted average of those two outcomes.

1. Player 2 plays R with probability q and L with probability 1-q.

2. Payoffs are read from Player 1's row L in the displayed matrix.

1. Indifference method for finding an opponent's mixing probability

1. Audio
   Observation

   At 01:10-01:19 the speaker says this will set the mixed strategies Nash equilibrium if both players choose those mixes.

2. Diagram
   Observation

   The board title reads 'Mixed Strategies Nash'.

1. The clip only derives Player 2's q; it does not restate Player 1's p within this segment.

If Player 2 chooses q so that $\Pi_1(R)=\Pi_1(L)$, then Player 1 is indifferent between R and L, and together with Player 1's corresponding mixing probability this yields the mixed-strategy Nash equilibrium of the displayed game.

1. The game is the displayed 2x2 normal-form game.

2. Player 2 mixes between R and L with probabilities q and 1-q.

3. The equilibrium concept is mixed-strategy Nash equilibrium.

For the displayed game, there exists a q making Player 1 indifferent; when both players use their equilibrium mixes, the profile is a mixed-strategy Nash equilibrium.

1. Formula
   Observation

   The speaker sequentially writes the equations for $\pi_2(R)$, $\pi_2(L)$, sets them equal, and solves for p.

1. Expression

   $$\pi_2(R) = p(1) + (1-p)(0)$$
   Explanation

   Calculate the expected payoff for Player 2 if they choose strategy R. Player 2 gets a payoff of 1 if Player 1 chooses R (probability p), and 0 if Player 1 chooses L (probability 1-p).

   Justification

   Definition of expected value.

   Shown in the video
2. Expression

   $$\pi_2(R) = p$$
   Explanation

   Simplify the expression for the expected payoff of Player 2 choosing R.

   Justification

   Algebraic simplification.

   Shown in the video
3. Expression

   $$\pi_2(L) = p(0) + (1-p)(2)$$
   Explanation

   Calculate the expected payoff for Player 2 if they choose strategy L. Player 2 gets a payoff of 0 if Player 1 chooses R (probability p), and 2 if Player 1 chooses L (probability 1-p).

   Justification

   Definition of expected value.

   Shown in the video
4. Expression

   $$\pi_2(L) = 2 - 2p$$
   Explanation

   Simplify the expression for the expected payoff of Player 2 choosing L.

   Justification

   Algebraic expansion and simplification.

   Shown in the video
5. Expression

   $$p = 2 - 2p$$
   Explanation

   Set the expected payoffs equal to each other to make Player 2 indifferent between R and L.

   Justification

   Indifference principle.

   Shown in the video
6. Expression

   $$3p = 2$$
   Explanation

   Add 2p to both sides of the equation.

   Justification

   Algebraic manipulation.

   Shown in the video
7. Expression

   $$p = 2/3$$
   Explanation

   Divide both sides by 3 to solve for p.

   Justification

   Algebraic manipulation.

   Shown in the video

Player 1 must choose strategy R with probability $p = 2/3$ to make Player 2 indifferent between their strategies.

1. Diagram
   Observation

   The board shows $3-3q=4q$, then $3=7q$, then $q=3/7$ boxed.

2. Audio
   Observation

   At 00:48-00:57 the speaker says the two expressions must be set equal to each other.

1. Expression

   $$\Pi_1(R)=q(0)+(1-q)3=3-3q$$
   Explanation

   Compute Player 1's expected payoff from choosing R against Player 2's mixture.

   Justification

   Weighted-average expected payoff using the top-row payoffs 0 and 3.

   Shown in the video
2. Expression

   $$\Pi_1(L)=q(4)+(1-q)0=4q$$
   Explanation

   Compute Player 1's expected payoff from choosing L against Player 2's mixture.

   Justification

   Weighted-average expected payoff using the bottom-row payoffs 4 and 0.

   Shown in the video
3. Expression

   $$3-3q=4q$$
   Explanation

   Set the two expected payoffs equal so Player 1 is indifferent between R and L.

   Justification

   Indifference condition for a mixed-strategy equilibrium component.

   Shown in the video
4. Expression

   $$3=7q$$
   Explanation

   Move the -3q term to the right-hand side and combine like terms.

   Justification

   Algebraic rearrangement of a linear equation.

   Shown in the video
5. Expression

   $$q=\frac{3}{7}$$
   Explanation

   Divide both sides by 7 to isolate q.

   Justification

   Solving a one-variable linear equation.

   Shown in the video

The value of q that makes Player 1 indifferent is $q=3/7$.

1. Diagram
   Observation

   The whiteboard displays a specific 2x2 payoff matrix used throughout the derivation.

Given a 2-player game with the payoff matrix [[0, 1], [3, 0]] for Player 1's row R and [[4, 0], [0, 2]] for row L, find the probability p that Player 1 should choose row R to make Player 2 indifferent between their columns.

1. Payoff matrix: Row R yields (0,1) if Col R, (3,0) if Col L. Row L yields (4,0) if Col R, (0,2) if Col L.

2. Player 1 chooses R with probability p, L with probability 1-p.

3. Player 2 chooses R with probability q, L with probability 1-q.

Determine the value of p.

1. Expression

   $$\pi_2(R) = p(1) + (1-p)(0) = p$$
   Explanation

   Compute Player 2's expected payoff for choosing column R.

   Justification

   Expected value formula applied to Player 2's payoffs in the first column.

   Shown in the video
2. Expression

   $$\pi_2(L) = p(0) + (1-p)(2) = 2 - 2p$$
   Explanation

   Compute Player 2's expected payoff for choosing column L.

   Justification

   Expected value formula applied to Player 2's payoffs in the second column.

   Shown in the video
3. Expression

   $$p = 2 - 2p \implies 3p = 2 \implies p = 2/3$$
   Explanation

   Equate the expected payoffs and solve for p.

   Justification

   Indifference principle requires $\pi_2(R) = \pi_2(L)$.

   Shown in the video

$p = 2/3$

Substituting $p = 2/3$ back into the expected payoff equations gives $\pi_2(R) = 2/3$ and $\pi_2(L) = 2 - 2(2/3) = 2 - 4/3 = 2/3$, confirming Player 2 is indifferent.

1. Diagram
   Observation

   The full worked example is visible on the board: matrix, two payoff formulas, equality, and boxed $q=3/7$.

2. Audio
   Observation

   The speaker narrates the setup, equality step, and interpretation of $q=3/7$.

1. The clip references Player 1's earlier solution but does not show its derivation in this segment.

Given the displayed 2x2 payoff matrix, determine the probability q with which Player 2 should choose R so that Player 1 is indifferent between R and L.

1. Player 1 has strategies R and L.

2. Player 2 has strategies R and L.

3. Payoff matrix entries are (0,1), (3,0), (4,0), (0,2).

4. Player 2 chooses R with probability q and L with probability 1-q.

Find q such that Player 1's expected payoff from R equals Player 1's expected payoff from L.

1. Expression

   $$\Pi_1(R)=q(0)+(1-q)3=3-3q$$
   Explanation

   Use the top-row payoffs for Player 1 to form the expected payoff from R.

   Justification

   Expected value under Player 2's mixed strategy.

   Shown in the video
2. Expression

   $$\Pi_1(L)=q(4)+(1-q)0=4q$$
   Explanation

   Use the bottom-row payoffs for Player 1 to form the expected payoff from L.

   Justification

   Expected value under Player 2's mixed strategy.

   Shown in the video
3. Expression

   $$3-3q=4q$$
   Explanation

   Equate the two expected payoffs to impose indifference.

   Justification

   Mixed-strategy equilibrium requires the mixing player to be indifferent across used pure strategies.

   Shown in the video
4. Expression

   $$3=7q$$
   Explanation

   Add 3q to both sides.

   Justification

   Linear equation simplification.

   Shown in the video
5. Expression

   $$q=\frac{3}{7}$$
   Explanation

   Divide by 7.

   Justification

   Isolating the unknown probability.

   Shown in the video

$q=\frac{3}{7}$

Substitute back: $\Pi_1(R)=3-3(3/7)=12/7$ and $\Pi_1(L)=4(3/7)=12/7$, so the two payoffs are equal.

1. Diagram
   Observation

   The video shows a whiteboard with a pre-drawn 2x2 payoff matrix, labels for players and strategies, and probabilities p, 1-p, q, 1-q.

1. 2x2 payoff matrix

2. Player 1 strategies (R, L)

3. Player 2 strategies (R, L)

4. Probabilities (p, 1-p, q, 1-q)

5. Expected payoff equations

1. The speaker writes the equations for $\pi_2(R)$ and $\pi_2(L)$ below the matrix.

2. The speaker sets the equations equal and solves for p, writing the steps on the board.

1. The initial payoff matrix and its values remain unchanged throughout the clip.

The visual layout organizes the game's parameters and the step-by-step mathematical derivation of the mixed strategy equilibrium, making the logical flow clear.

1. Diagram
   Observation

   The presenter erases the previous 'Player 1's choice of p' work and writes a new section headed 'Player 2's choice of q:'.

2. Animation
   Observation

   Across the clip the board accumulates $\Pi_1(R)$, $\Pi_1(L)$, the equality $3-3q=4q$, then $3=7q$, and finally the boxed result $q=3/7$.

1. 2x2 payoff matrix

2. heading 'Player 2's choice of q:'

3. two expected-payoff formulas

4. equation chain ending in boxed $q=3/7$

1. The earlier p-related equations are removed.

2. A new heading for Player 2's q is written.

3. Two payoff expressions are added one after the other.

4. The indifference equation and its algebraic simplification are written.

5. The final answer is boxed.

1. The payoff matrix itself remains unchanged throughout the clip.

2. Player 1's strategies remain labeled R and L, and Player 2's remain labeled R and L.

The visual sequence shows the standard mixed-strategy calculation: hold the game fixed, switch from solving one player's mixing probability to the other player's, derive expected payoffs, impose indifference, and solve the resulting linear equation.

1. Audio
   Observation

   The speaker notes, 'Now that is a little bit counterintuitive, but once again in this video I'm just showing you the math, I'm not showing you the intuition.'

Students might intuitively think a player should choose probabilities to maximize their own expected payoff directly, rather than to make the opponent indifferent.

In a mixed strategy Nash equilibrium, a player's mixing probabilities are chosen specifically to make the *other* player indifferent between their pure strategies, ensuring the opponent has no incentive to deviate from their own mixed strategy.

1. Audio
   Observation

   At 01:00-01:09 the speaker explicitly interprets q as the probability that Player 2 goes right.

2. Diagram
   Observation

   The equations pair q with the R-column payoffs and (1-q) with the L-column payoffs.

One may mistakenly attach q to Player 2's left action or to the wrong column payoff.

In this worked example, q multiplies the payoffs in Player 2's R column, so q is the probability of Player 2 choosing R, while (1-q) is the probability of choosing L.

1. Audio
   Observation

   At 00:08-00:15 the speaker says they need to make Player 1 indifferent when solving for Player 2's q.

A common error is to use Player 2's own payoffs when solving for Player 2's q.

To find Player 2's equilibrium mixing probability, one equalizes Player 1's expected payoffs across Player 1's pure strategies, because Player 2's mixture must make Player 1 willing to mix.

1. Audio
   Observation

   The speaker explicitly links the indifference principle to setting up and solving an algebraic equation.

The derivation calculates the exact probability p by applying the indifference principle, setting the expected payoffs equal and solving the resulting equation.

1. Diagram
   Observation

   The board first computes $\Pi_1(R)$ and $\Pi_1(L)$, then sets them equal.

The indifference method is applied by first computing the two expected-payoff expressions and then equating them.

1. Diagram
   Observation

   Both $\Pi_1(R)$ and $\Pi_1(L)$ feed into the equation $3-3q=4q$.

The second expected-payoff formula supplies the other side of the indifference equation used in the method.

1. Diagram
   Observation

   The example and derivation use the same board work ending in $q=3/7$.

The worked example is solved by the derivation that sets the two expected payoffs equal and simplifies to $q=3/7$.

1. Audio
   Observation

   The entire segment focuses on finding the value of p for Player 1.

1. Indifference Principle in Mixed Nash Equilibrium
2. Derivation of Player 1's Mixed Strategy Probability p
3. Finding Mixed Strategy Probability for Player 1

1. Formula
   Observation

   The speaker writes out the formulas for $\pi_2(R)$ and $\pi_2(L)$.

1. Derivation of Player 1's Mixed Strategy Probability p
2. Finding Mixed Strategy Probability for Player 1

1. Audio
   Observation

   The whole clip is devoted to finding Player 2's q.

1. Indifference method for finding an opponent's mixing probability
2. Expected payoff to Player 1 from playing R
3. Expected payoff to Player 1 from playing L
4. Solving the indifference equation for q
5. Finding Player 2's mixing probability in a 2x2 game

1. Audio
   Observation

   At 00:48-00:57 the speaker says the two things must be set equal to each other to make Player 1 indifferent.

1. Indifference method for finding an opponent's mixing probability
2. Equating Player 1's pure-strategy payoffs identifies Player 2's equilibrium mixing probability
3. Solving the indifference equation for q

1. Audio
   Observation

   At 00:08-00:15 the speaker says Player 2's q is chosen to make Player 1 indifferent.

1. Indifference method for finding an opponent's mixing probability
2. Thinking each player's own payoff is used to solve that player's mixing probability

1. Audio
   Observation

   At 01:00-01:09 the speaker interprets q as the probability that Player 2 goes right.

1. q
2. Finding Player 2's mixing probability in a 2x2 game
3. Equating Player 1's pure-strategy payoffs identifies Player 2's equilibrium mixing probability
4. Confusing which outcome receives probability q