Solving for Mixed Strategies Nash Equilibrium (Example)

1. Diagram
   Observation

   The payoff matrix header shows P above Hawk and 1-P above Dove.

2. Audio
   Observation

   The instructor says the other player mixes between hawk and dove with a probability of p going hawk and a probability of 1 minus p going dove.

P

Probability that the opponent plays Hawk.

$0 \le P \le 1$

1. Diagram
   Observation

   The payoff matrix header shows 1-P above Dove.

2. Audio
   Observation

   The instructor refers to the probability of the other player going dove as 1 minus p.

1-P

Probability that the opponent plays Dove.

$0 \le 1-P \le 1$

1. Diagram
   Observation

   Hawk labels the first row and the first column of the payoff matrix.

2. Audio
   Observation

   The instructor repeatedly names the hawk strategy.

Hawk

One pure strategy in the Game of Chicken payoff matrix.

1. Diagram
   Observation

   Dove labels the second row and the second column of the payoff matrix.

2. Audio
   Observation

   The instructor repeatedly names the dove strategy.

Dove

One pure strategy in the Game of Chicken payoff matrix.

1. Diagram
   Observation

   The whiteboard shows a 2x2 matrix with entries (-1,-1), (4,0), (0,4), and (1,1).

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

Payoff matrix for the Game of Chicken, with row player payoffs listed first and column player payoffs second.

1. Formula
   Observation

   The instructor writes $\Pi_1(H)$ on the board.

2. Audio
   Observation

   She says she is finding the payoff to player one from choosing hawk.

$\Pi_1(H)$

Expected payoff to player 1 when player 1 chooses Hawk against the opponent's mixed strategy.

1. Formula
   Observation

   The instructor writes $\Pi_1(D)$ on the board.

2. Audio
   Observation

   She says similarly the payoff to player one of going dove is going to be equal to ...

$\Pi_1(D)$

Expected payoff to player 1 when player 1 chooses Dove against the opponent's mixed strategy.

1. Diagram
   Observation

   The whiteboard shows 'p' above the Hawk column and '1-p' above the Dove column in the payoff matrix.

2. Audio
   Observation

   The speaker refers to 'p' as the other player's mix.

p

Probability that Player 2 chooses Hawk.

[0, 1]

1. Diagram
   Observation

   The speaker writes 'q' next to the Hawk row and '1-q' next to the Dove row.

2. Audio
   Observation

   The speaker states, 'I'm going to let player one mix with a probability of q and one minus q.'

q

Probability that Player 1 chooses Hawk.

[0, 1]

1. Diagram
   Observation

   The whiteboard displays the equation '$\Pi_1(H) = p(-1) + (1-p)(4) = -p + 4 - 4p = -5p + 4$'.

$\Pi_1(H)$

Expected payoff for Player 1 when choosing the pure strategy Hawk.

Real numbers

1. Diagram
   Observation

   The whiteboard displays the equation '$\Pi_1(D) = p(0) + (1-p)(1) = 1 - p$'.

$\Pi_1(D)$

Expected payoff for Player 1 when choosing the pure strategy Dove.

Real numbers

1. Diagram
   Observation

   The speaker writes 'max_q' followed by an expression for the total expected payoff function.

2. Audio
   Observation

   The speaker says, 'we maximize player one's objective function by choosing q'.

$\max_q$

Maximization operator with respect to the variable q.

Real numbers

1. Diagram
   Observation

   The board title reads Stable & Unstable Nash and the matrix is labeled Game of Chicken.

2. Audio
   Observation

   The instructor says this particular game of chicken.

The clip uses a two-strategy symmetric game called the Game of Chicken. The strategies are Hawk and Dove, and the displayed payoff matrix gives each outcome pair as (row player payoff, column player payoff).

1. Two players.

2. Each player has strategies Hawk and Dove.

3. Payoffs are written as ordered pairs with the row player first.

1. Audio
   Observation

   The instructor says she is going to find the mixed strategies Nash equilibrium of this particular game of chicken.

2. Diagram
   Observation

   The matrix headers use P and 1-P to represent mixing probabilities.

In this example, a mixed strategy is represented by assigning probability P to Hawk and probability 1-P to Dove. The instructor seeks the value of P that makes the opponent indifferent between their own pure strategies.

1. The opponent randomizes between Hawk and Dove.

2. Probabilities must sum to 1.

1. Game of Chicken payoff matrix

1. Audio
   Observation

   The instructor says one strategy is to choose a probability of going hawk that makes the other player indifferent between their strategies.

2. Audio
   Observation

   Later she says player one will be indifferent when the payoff from going hawk is equal to the payoff of going dove.

The method shown is to compute the expected payoff of each pure strategy against the opponent's mix, then set those expected payoffs equal. Solving that equality gives the mixing probability that makes the opponent indifferent.

1. Use when looking for a mixed-strategy equilibrium supported by the listed pure strategies.

2. Compare the same player's expected payoffs across pure strategies.

1. Mixed strategy Nash equilibrium in this example

1. Audio
   Observation

   The instructor says it doesn't actually matter here because this is a symmetric game, like the players have the same payoffs if you flip it.

The instructor states that the game is symmetric, meaning the players have the same payoff structure under interchange of roles. This is why she can analyze player 1's payoffs without separately redoing player 2's case in this clip.

1. The payoff structure is unchanged when the players' roles are flipped.

1. Game of Chicken payoff matrix

1. Audio
   Observation

   The speaker explains two methods for finding mixed strategy Nash equilibria: equating expected payoffs of pure strategies and using optimization with first-order conditions.

2. Diagram
   Observation

   The whiteboard shows the Game of Chicken payoff matrix and the step-by-step derivation of the mixed strategy probability $p = 3/4$ using both methods.

A mixed strategy Nash equilibrium can be found by ensuring a player is indifferent between their pure strategies. This can be done by setting the expected payoffs of the pure strategies equal to each other, or by setting up an optimization problem for a player's expected payoff with respect to their mixing probability and taking the first-order condition. In a symmetric game, solving for one player's optimal mixing probability gives the other player's as well.

1. The game must have at least one mixed strategy Nash equilibrium.

2. For the optimization method, the expected payoff function must be differentiable with respect to the mixing probability.

1. Diagram
   Observation

   The whiteboard shows calculations like '$\Pi_1(H) = p(-1) + (1-p)(4)$' and the full expected payoff function for Player 1.

The expected payoff for a player choosing a specific pure strategy against an opponent's mixed strategy is calculated by multiplying the payoff of each possible outcome by the probability of the opponent choosing that corresponding pure strategy, and summing the results. The total expected payoff for a player using a mixed strategy is the sum of the expected payoffs of each pure strategy weighted by their respective probabilities.

1. The probabilities of the opponent's strategies must sum to 1.

1. Audio
   Observation

   The instructor says player one will be indifferent when the payoff from going hawk is equal to the payoff of going dove.

2. Formula
   Observation

   She then sets -5p+$4 = 1-p$ and solves.

For the mixed strategy being solved here, player 1 is indifferent between Hawk and Dove exactly when $\Pi_1(H)=\Pi_1(D)$.

1. Player 2 mixes with probabilities P and 1-P over Hawk and Dove.

2. The relevant payoffs are the expected payoffs to player 1 from each pure strategy.

For the mixing probability P under consideration in this example.

1. Audio
   Observation

   The instructor says when we do that we get p equals three fourths is going to be the mixed strategies Nash equilibrium.

2. Formula
   Observation

   The board shows the boxed result $p=3/4$.

Solving the indifference equation for this payoff matrix gives $P=\frac{3}{4}$.

1. Use the displayed Game of Chicken payoff matrix.

2. Set $\Pi_1(H)=\Pi_1(D)$.

For this specific 2x2 Game of Chicken example.

1. Audio
   Observation

   The speaker states, 'it would look exactly the same since it's a symmetric problem.'

In the Game of Chicken, the calculation for Player 2's expected payoff from going Hawk is identical to Player 1's because the game is symmetric.

1. The game is a symmetric game.

Universal for all symmetric games.

1. Audio
   Observation

   The speaker notes, 'what popped out of it had nothing to do with q, it had to do with p, the other player's mix. And that's how mixed strategies Nash equilibrium works.'

When maximizing a player's expected payoff with respect to their own mixing probability (q) to find a mixed strategy Nash equilibrium, the first-order condition will yield an equation that determines the *opponent's* mixing probability (p), not the player's own.

1. The game has a mixed strategy Nash equilibrium.

2. The expected payoff function is linear in the player's own mixing probability.

Universal for 2x2 games with mixed strategy Nash equilibria.

1. Formula
   Observation

   The instructor writes $\Pi_1(H)=p(-1)+(1-p)(4)=-p+4-4p=-5p+4$.

2. Audio
   Observation

   She explains each term as probability times payoff for encountering hawk or dove.

1. Expression

   $$\Pi_1(H)=p(-1)+(1-p)(4)$$
   Explanation

   The expected payoff from choosing Hawk is the weighted sum of the two possible opponent actions.

   Justification

   Definition of expected payoff using the opponent's mixing probabilities P and 1-P.

   Shown in the video
2. Expression

   $$=-p+4-4p$$
   Explanation

   Distribute the multiplication over the terms.

   Justification

   Algebraic expansion.

   Shown in the video
3. Expression

   $$=-5p+4$$
   Explanation

   Combine like terms in p.

   Justification

   Algebraic simplification.

   Shown in the video

$\Pi_1(H)=-5p+4$.

1. Formula
   Observation

   The instructor writes $\Pi_1(D)=p(0)+(1-p)(1)=1-p$.

2. Audio
   Observation

   She says the expected payoff is the probability of encountering a hawk times the payoff if you encounter a hawk plus the probability you encounter a dove times the payoff ... is one.

1. Expression

   $$\Pi_1(D)=p(0)+(1-p)(1)$$
   Explanation

   The expected payoff from choosing Dove is the weighted sum of the two possible opponent actions.

   Justification

   Definition of expected payoff using the opponent's mixing probabilities.

   Shown in the video
2. Expression

   $$=1-p$$
   Explanation

   The first term vanishes and the second term simplifies.

   Justification

   Arithmetic simplification.

   Shown in the video

$\Pi_1(D)=1-p$.

1. Formula
   Observation

   The board shows -5p+$4=1-p$, then +5p-1, then $3=4p$, then $p=3/4$ boxed.

2. Audio
   Observation

   The instructor says set these two things equal to one another and solve for p, then states p equals three fourths.

1. Expression

   $$-5p+4=1-p$$
   Explanation

   Set the expected payoff from Hawk equal to the expected payoff from Dove.

   Justification

   Indifference condition for a mixed strategy equilibrium.

   Shown in the video
2. Expression

   $$+5p-1$$
   Explanation

   Move terms to isolate constants and coefficients of p.

   Justification

   Algebraic rearrangement.

   Shown in the video
3. Expression

   $$3=4p$$
   Explanation

   After rearrangement, the constant terms give 3 and the p terms give 4p.

   Justification

   Algebraic simplification.

   Shown in the video
4. Expression

   $$p=\frac{3}{4}$$
   Explanation

   Divide both sides by 4.

   Justification

   Solving a linear equation.

   Shown in the video

The mixing probability satisfying indifference is $p=\frac{3}{4}$.

1. Diagram
   Observation

   The whiteboard shows the equations '$\Pi_1(H) = -5p + 4$' and '$\Pi_1(D) = 1 - p$', followed by '-5p + $4 = 1 - p \Rightarrow 3 = 4p \Rightarrow p = 3/4$'.

1. Expression

   $$\Pi_1(H) = p(-1) + (1-p)(4) = -5p + 4$$
   Explanation

   Calculate the expected payoff for Player 1 if they choose Hawk, given Player 2 mixes with probability p for Hawk.

   Justification

   Definition of expected value.

   Shown in the video
2. Expression

   $$\Pi_1(D) = p(0) + (1-p)(1) = 1 - p$$
   Explanation

   Calculate the expected payoff for Player 1 if they choose Dove, given Player 2 mixes with probability p for Hawk.

   Justification

   Definition of expected value.

   Shown in the video
3. Expression

   $$-5p + 4 = 1 - p$$
   Explanation

   Set the two expected payoffs equal to each other to find the indifference point for Player 1.

   Justification

   Condition for a mixed strategy Nash equilibrium (player must be indifferent between pure strategies).

   Shown in the video
4. Expression

   $$3 = 4p \Rightarrow p = 3/4$$
   Explanation

   Solve the resulting linear equation for p.

   Justification

   Algebraic manipulation.

   Shown in the video

Player 2 must play Hawk with probability $p = 3/4$ to make Player 1 indifferent between Hawk and Dove.

1. Diagram
   Observation

   The whiteboard shows the setup of the maximization problem '$\max_q q[p(-1) + (1-p)4] + (1-q)[p(0) + (1-p)1]$', its simplification to '$\max_q -4$qp + $3q + 1 - p$', and the first-order condition '$\frac{\partial \pi}{\partial q} = -4p + 3 = 0 \Rightarrow p = 3/4$'.

1. Expression

   $$\max_q \left( q[p(-1) + (1-p)4] + (1-q)[p(0) + (1-p)1] \right)$$
   Explanation

   Set up the optimization problem to maximize Player 1's total expected payoff with respect to their mixing probability q.

   Justification

   Definition of expected payoff for a mixed strategy.

   Shown in the video
2. Expression

   $$\max_q (-4qp + 3q + 1 - p)$$
   Explanation

   Simplify the expected payoff expression by expanding and combining like terms.

   Justification

   Algebraic simplification.

   Shown in the video
3. Expression

   $$\frac{\partial \pi}{\partial q} = -4p + 3$$
   Explanation

   Take the partial derivative of the simplified expected payoff function with respect to q.

   Justification

   Rules of differentiation (power rule, treating p as a constant).

   Shown in the video
4. Expression

   $$-4p + 3 = 0$$
   Explanation

   Set the first-order condition (the derivative) equal to zero to find the critical point.

   Justification

   Necessary condition for an interior maximum in optimization.

   Shown in the video
5. Expression

   $$p = 3/4$$
   Explanation

   Solve the resulting equation for p.

   Justification

   Algebraic manipulation.

   Shown in the video

The first-order condition yields $p = 3/4$, which is the probability Player 2 must play Hawk to make Player 1's expected payoff maximized (and thus indifferent between pure strategies).

1. Diagram
   Observation

   The whole clip works on the displayed Game of Chicken matrix.

2. Audio
   Observation

   The instructor explicitly frames the video as solving for the mixed strategies Nash equilibrium of this game.

Find the mixed strategy probability P that makes the opponent indifferent in the displayed Game of Chicken.

1. Strategies: Hawk and Dove.

2. Payoff matrix: Hawk/Hawk=(-1,-1), Hawk/Dove=(4,0), Dove/Hawk=(0,4), Dove/Dove=(1,1).

3. Opponent mixes with probability P on Hawk and 1-P on Dove.

Compute the mixed-strategy equilibrium probability P.

1. Expression

   $$\Pi_1(H)=p(-1)+(1-p)(4)$$
   Explanation

   Write the expected payoff to player 1 from choosing Hawk.

   Justification

   Expected value over the opponent's two pure strategies.

   Shown in the video
2. Expression

   $$\Pi_1(H)=-5p+4$$
   Explanation

   Simplify the Hawk payoff expression.

   Justification

   Algebra.

   Shown in the video
3. Expression

   $$\Pi_1(D)=p(0)+(1-p)(1)$$
   Explanation

   Write the expected payoff to player 1 from choosing Dove.

   Justification

   Expected value over the opponent's two pure strategies.

   Shown in the video
4. Expression

   $$\Pi_1(D)=1-p$$
   Explanation

   Simplify the Dove payoff expression.

   Justification

   Algebra.

   Shown in the video
5. Expression

   $$-5p+4=1-p$$
   Explanation

   Set the two expected payoffs equal.

   Justification

   Indifference condition for mixed equilibrium.

   Shown in the video
6. Expression

   $$p=\frac{3}{4}$$
   Explanation

   Solve the resulting linear equation.

   Justification

   Algebraic solution.

   Shown in the video

$P=\frac{3}{4}$.

Substitute $P=3/4$ into the two payoff formulas: $\Pi_1(H)=-5(3/4)+4=1/4$ and $\Pi_1(D)=1-3/4=1/4$, so the payoffs are equal.

1. Diagram
   Observation

   The whiteboard displays the 'Game of Chicken' payoff matrix with Hawk and Dove strategies and payoffs (-1,-1), (4,0), (0,4), (1,1).

Find the mixed strategy Nash equilibrium for the Game of Chicken.

1. Payoff matrix for Game of Chicken: Hawk/Hawk = (-1,-1), Hawk/Dove = (4,0), Dove/Hawk = (0,4), Dove/Dove = (1,1).

2. Player 2 plays Hawk with probability p and Dove with probability 1-p.

3. Player 1 plays Hawk with probability q and Dove with probability 1-q.

Determine the probability p (and by symmetry, q) that constitutes a mixed strategy Nash equilibrium.

1. Expression

   $$\Pi_1(H) = -5p + 4, \Pi_1(D) = 1 - p$$
   Explanation

   Calculate Player 1's expected payoffs for pure strategies Hawk and Dove.

   Justification

   Definition of expected value.

   Shown in the video
2. Expression

   $$-5p + 4 = 1 - p \Rightarrow p = 3/4$$
   Explanation

   Equate the expected payoffs and solve for p.

   Justification

   Indifference condition for mixed strategy Nash equilibrium.

   Shown in the video
3. Expression

   $$\max_q (-4qp + 3q + 1 - p)$$
   Explanation

   Set up and simplify Player 1's total expected payoff function.

   Justification

   Definition of expected payoff for a mixed strategy.

   Shown in the video
4. Expression

   $$\frac{\partial \pi}{\partial q} = -4p + 3 = 0 \Rightarrow p = 3/4$$
   Explanation

   Take the first-order condition with respect to q and solve for p.

   Justification

   Optimization condition for Nash equilibrium.

   Shown in the video

The mixed strategy Nash equilibrium involves each player playing Hawk with probability $3/4$ and Dove with probability $1/4$.

Both methods (equating expected payoffs and using first-order conditions) yield the same result, $p = 3/4$.

1. Diagram
   Observation

   A whiteboard shows the title Stable & Unstable Nash, a 2x2 payoff matrix labeled Game of Chicken, and later handwritten equations beneath it.

1. Title text: Stable & Unstable Nash

2. 2x2 payoff matrix

3. Row labels Hawk and Dove

4. Column headers P Hawk and 1-P Dove

5. Side label Game of Chicken

6. Handwritten equations below the matrix

1. At the start, only the title and payoff matrix are visible.

2. Around 42 seconds, the instructor begins writing $\Pi_1(H)$.

3. Around 94 seconds, she adds $\Pi_1(D)$.

4. Around 128 seconds, she writes the equality -5p+$4=1-p$ and solves it.

5. By the end, the boxed result $p=3/4$ is visible.

1. The payoff matrix values remain unchanged throughout the clip.

2. The strategy labels Hawk and Dove remain fixed.

3. The column headers P and 1-P remain fixed.

The visual progression mirrors the algebraic method: first identify the game, then compute each pure-strategy expected payoff, then impose indifference and solve for the mixing probability.

1. Animation
   Observation

   The speaker erases previous calculations and writes new equations for the optimization method on the whiteboard.

1. Whiteboard

2. Marker

3. Speaker

1. Previous equations are erased.

2. New variables q and 1-q are written next to the payoff matrix rows.

3. The maximization problem is written out step-by-step.

4. The first-order condition is derived and solved.

1. The Game of Chicken payoff matrix remains visible at the top of the board.

The visual progression demonstrates the alternative method for finding the mixed strategy Nash equilibrium using calculus-based optimization.

1. Audio
   Observation

   The instructor says it doesn't actually matter here because this is a symmetric game, like the players have the same payoffs if you flip it.

One might think the calculation must be repeated separately for both players even in a symmetric game.

The instructor explicitly notes that because the game is symmetric, the same payoff logic applies after flipping the players, so analyzing player 1 suffices for this example.

1. Audio
   Observation

   The speaker explicitly points out the counter-intuitive result: 'you notice it's a little bit weird that we set up an optimization problem trying to decide what we as player one should choose for our mix, that's q, and what popped out of it had nothing to do with q, it had to do with p, the other player's mix.'

Students might expect that maximizing their own payoff with respect to their own mixing probability (q) would directly yield their optimal q.

In a mixed strategy Nash equilibrium, a player is indifferent between their pure strategies. The first-order condition of the expected payoff function with respect to a player's own mixing probability actually reveals the *opponent's* required mixing probability to maintain this indifference.

1. Audio
   Observation

   The instructor introduces the indifference method as a way to find the mixed strategies Nash equilibrium.

The indifference method is the computational procedure used to find the mixed strategy Nash equilibrium in this example.

1. Audio
   Observation

   The instructor says this is a symmetric game while referring to the displayed payoff matrix.

The displayed Game of Chicken is treated as an instance of a symmetric game, which justifies focusing on one player's payoffs.

1. Formula
   Observation

   The board equates the simplified expressions for $\Pi_1(H)$ and $\Pi_1(D)$.

The indifference claim depends on first deriving the expected payoff expressions for Hawk and Dove.

1. Formula
   Observation

   The board equates -5p+4 and 1-p.

The Dove expected payoff derivation supplies the right-hand side of the indifference equation.

1. Formula
   Observation

   The equality is solved to obtain the boxed $p=3/4$.

The equilibrium probability follows directly from solving the indifference equation.

1. Audio
   Observation

   The speaker presents two different methods and shows they lead to the same answer, stating 'You can use either method depending on what you're most comfortable with'.

Equating expected payoffs of pure strategies and using optimization with first-order conditions are equivalent methods for finding mixed strategy Nash equilibria in 2x2 games.

1. Diagram
   Observation

   The derivation explicitly builds the expected payoff function and then applies the first-order condition to it.

The concept of expected payoff is applied to construct the objective function for the optimization method of finding a mixed strategy Nash equilibrium.

1. Diagram
   Observation

   The board labels the matrix Game of Chicken.

1. Game of Chicken payoff matrix

1. Audio
   Observation

   The instructor explains choosing a probability that makes the other player indifferent.

1. Indifference method for mixed equilibrium
2. Mixed strategy Nash equilibrium in this example

1. Audio
   Observation

   The instructor says it does not matter because the game is symmetric.

1. Symmetric game remark

1. Audio
   Observation

   The instructor says player one will be indifferent when the payoff from going hawk is equal to the payoff of going dove.

1. Indifference condition for mixed equilibrium
2. Indifference method for mixed equilibrium

1. Formula
   Observation

   The final boxed result is $p=3/4$.

1. Computed mixed equilibrium probability
2. Worked example: mixed strategy Nash equilibrium in Game of Chicken

1. Audio
   Observation

   The entire video is dedicated to explaining how to find a mixed strategy Nash equilibrium.

1. Finding Mixed Strategy Nash Equilibrium

1. Audio
   Observation

   The speaker addresses the confusion about why the first-order condition with respect to q yields p.

1. Finding Mixed Strategy Nash Equilibrium
2. Misinterpreting the Result of the First Order Condition

1. Diagram
   Observation

   The whiteboard clearly labels the matrix as 'Game of Chicken'.

1. Mixed Strategy Nash Equilibrium in Game of Chicken