Ashley Hodgson · YouTube · 5:17
This 165-second whiteboard lesson introduces Nash equilibrium through an informal definition and a fully worked 2x2 game. The speaker first describes equilibrium as a “no regrets” situation and then rephrases it as “best response to best response,” writing that phrase on the board. She next explains how to read a payoff matrix using a Harry-Hermione example, where Harry is player 1 with red first-entry payoffs and Hermione is player 2 with purple second-entry payoffs. The method is demonstrated step by step: hold the opponent’s strategy fixed, compare only your own payoffs, and circle the best response. Harry’s comparisons show that Down beats Up in both columns, so Down is a dominant strategy. Hermione’s comparisons show that Left beats Right in both rows. The cell (6,3) receives both players’ circles, and the speaker identifies it as the Nash equilibrium. She then emphasizes that this equilibrium is not the cell both players would prefer, calling the example a prisoner’s dilemma. In the last seconds she begins moving to another matrix involving Elizabeth and Mr. Darcy, but the clip ends before that second example is solved. This 152-second whiteboard lesson teaches how to find pure-strategy Nash equilibria in 2x2 payoff matrices by identifying each player's best responses. The presenter first works the Elizabeth vs. Mr. Darcy game and finds two equilibria, (Up, Left) and (Down, Right). She then gives a practical tip for uncolored matrices—underline Player 1's payoffs—and applies it to the Ron vs. Neville game, concluding there is no pure Nash equilibrium. Finally, she solves the Emma vs. Mr. Knightley game, where only (Down, Right) = (8,7) is doubly circled. The clip closes by defining Nash equilibrium as a "no regrets" outcome: a best response to a best response.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
The clip opens with the question “What is Nash equilibrium?” and immediately answers it in plain language: it is a “no regrets equilibrium.” The intended point is that once all choices are fixed, no player wishes they had chosen differently given the other player’s choice.
The presenter then gives an equivalent formulation and writes it on the board: “Best response to best response.” This shifts the idea from a psychological description (“no regrets”) to a strategic one: each player’s action must be optimal relative to the other player’s action.
Before solving an example, the speaker adds a caution: equilibrium does not automatically mean the best collective outcome. This sets up the later contrast between the equilibrium cell and another cell that both players would prefer.
The first worked game is a 2x2 matrix for Harry and Hermione. The board convention is explained explicitly: Harry is player 1, his payoffs are the first entries and are written in red, and his strategies are the rows Up and Down. Hermione is player 2, with the second entries in purple and strategies Left and Right across the columns.
To find Harry’s best responses, the method is to freeze Hermione’s choice and compare only Harry’s own payoffs. In the Left column, Harry compares 4 from Up with 6 from Down and prefers 6. In the Right column, he compares 7 from Up with 9 from Down and prefers 9. These comparisons are marked directly on the matrix by circling the winning red payoffs.
Because Down wins in both columns, the speaker names this a dominant strategy for Harry: his best response is Down no matter what Hermione does. This is a stronger statement than merely identifying one best reply; it says the same row is optimal against every opponent column.
The perspective then flips to Hermione. Now Harry’s row is held fixed and Hermione compares only her second-entry payoffs. In the Up row she prefers 8 to 4, so Left is circled. In the Down row she prefers 3 to 2, so Left is circled again. The visual procedure mirrors the earlier one, but now along rows instead of columns.
With both players’ best responses marked, the presenter states the decision rule: if two circles appear in the same cell, that cell is a Nash equilibrium. In this matrix the doubly circled cell is (6,3), corresponding to Harry choosing Down and Hermione choosing Left.
The speaker then highlights why this example matters: although (6,3) is the equilibrium, both players would prefer another cell. This demonstrates the central warning from the introduction—Nash equilibrium is about mutual best response, not about achieving the best joint outcome. The example is labeled a prisoner’s dilemma, though the clip does not pause to define that term further.
In the final seconds, the presenter moves to the next matrix and begins the same procedure from “Elizabeth’s perspective,” checking the opponent’s strategies before comparing her own payoffs. The second example is only introduced here; no solution is completed within this segment.
The clip opens on a whiteboard titled "Nash Equilibrium" with several 2x2 payoff matrices. In the top-right example, Elizabeth is the row player with strategies Up and Down, and Mr. Darcy is the column player with strategies Left and Right. The matrix entries are (Up, Left) = (5,9), (Up, Right) = (2,6), (Down, Left) = (3,5), and (Down, Right) = (4,7).
The presenter begins by fixing Mr. Darcy's choice and asking what Elizabeth would prefer. If Darcy plays Left, Elizabeth compares 5 and 3 and chooses 5; if Darcy plays Right, she compares 4 and 2 and chooses 4. These are Elizabeth's best responses, so the corresponding first payoffs are circled.
Next she flips perspective to Mr. Darcy as Player 2. Holding Elizabeth's row fixed, he compares the second payoffs: if Elizabeth plays Up, he prefers 9 to 6; if she plays Down, he prefers 7 to 5. Those second payoffs are circled as well.
Because two cells now contain a circle for each player, the game has two pure-strategy Nash equilibria: (Up, Left) and (Down, Right). This illustrates that a small finite game can possess more than one equilibrium outcome.
The presenter then addresses a practical issue: what to do when the matrix is not color-coded by player. Her advice is to first underline all of Player 1's payoffs so that the comparison process stays organized and less error-prone.
She applies this to the bottom-left matrix, where Ron is Player 1 and Neville is Player 2. The entries are (Up, Left) = (9,4), (Up, Right) = (2,8), (Down, Left) = (5,6), and (Down, Right) = (7,3). She underlines 9, 2, 5, and 7 to mark Ron's payoffs.
Ron's best responses are found column by column: against Neville's Left he prefers 9 to 5, and against Right he prefers 7 to 2. Then Neville's best responses are found row by row using the non-underlined second payoffs: against Ron's Up he prefers 8 to 4, and against Down he prefers 6 to 3.
After all best responses are circled, no single cell contains two circles. Therefore this Ron-versus-Neville game has no pure-strategy Nash equilibrium, showing that mutual best response need not exist in every 2x2 game.
The final worked example is Emma versus Mr. Knightley, again with Emma as Player 1 and Mr. Knightley as Player 2. The matrix entries are (Up, Left) = (4,3), (Up, Right) = (5,4), (Down, Left) = (7,6), and (Down, Right) = (8,7).
Emma's best responses are computed first: if Mr. Knightley plays Left, she prefers 7 to 4; if he plays Right, she prefers 8 to 5. Thus her best response is Down in either column.
Then the perspective flips to Mr. Knightley's second payoffs: if Emma plays Up, he prefers 4 to 3; if Emma plays Down, he prefers 7 to 6. Thus his best response is Right in either row.
Only the bottom-right cell (Down, Right) = (8,7) ends up with two circles, one for each player. The presenter uses this to state the general rule from the board: a box with two circles is a best response to a best response, equivalently a "no regrets" strategy, and that is the Nash equilibrium condition being taught.
The video’s first definition is informal: a Nash equilibrium is an outcome where, after everyone has chosen, each player is satisfied with their own choice given what the others did. This is useful as an intuitive entry point before formal best-response language.
The same idea is restated more strategically: each player’s chosen action must be a best response to the other player’s chosen action. The board phrase “Best response to best response” summarizes the equilibrium condition used throughout the example.
A key caution is that Nash equilibrium does not mean the socially best or jointly preferred result. The later Harry-Hermione matrix is used to show an equilibrium that is worse for both players than another cell.
In the worked example, rows belong to the first player and columns to the second. Each cell contains an ordered pair of payoffs. Harry’s payoff is the first number (red), Hermione’s is the second number (purple).
To find one player’s best response, hold the opponent’s strategy fixed and compare only your own payoff entries. For the row player this means scanning down a column; for the column player it means scanning across a row.
A dominant strategy is a strategy that is a best response no matter what the opponent does. In the example, Harry prefers Down when Hermione plays Left and also prefers Down when Hermione plays Right, so Down is dominant for him.
Harry’s best responses are Down in both columns. Hermione’s best responses are Left in both rows. The only cell where both players’ best responses coincide is (6,3), so that cell is the Nash equilibrium.
After marking each player’s best responses separately, any cell containing both players’ marks is an equilibrium. Visually, “two circles in the same box” means both players are simultaneously playing best responses to each other.
The speaker notes that both players would prefer another cell than the equilibrium and identifies the example as a prisoner’s dilemma. The clip uses this to illustrate that mutual best response can still produce a collectively inferior outcome.
The final seconds begin a new matrix involving Elizabeth and Mr. Darcy, using the same method: inspect one player’s payoffs while checking the opponent’s strategies. The solution itself is not reached within this clip.
The video defines a Nash equilibrium as an outcome where each player is choosing a best response to the other player's strategy. The presenter summarizes this as a "no regrets" strategy: given the other player's choice, neither player wants to deviate.
To find pure Nash equilibria, fix one player's strategy and circle the other player's largest payoff in that row or column. Then switch players and repeat. Any cell with two circled payoffs, one belonging to each player, is a Nash equilibrium.
In the matrix (5,9), (2,6); (3,5), (4,7), Elizabeth's best responses are 5 and 4, while Mr. Darcy's are 9 and 7. The doubly circled cells are (Up, Left) and (Down, Right), so the game has two pure-strategy Nash equilibria.
When the matrix is not color-coded by player, underline all of Player 1's payoffs before searching for best responses. This prevents mixing up whose payoff is being compared during the circling process.
In the matrix (9,4), (2,8); (5,6), (7,3), Ron's best responses are 9 and 7, while Neville's are 8 and 6. No cell receives two circles, so there is no pure-strategy Nash equilibrium.
In the matrix (4,3), (5,4); (7,6), (8,7), Emma's best responses are 7 and 8, and Mr. Knightley's are 4 and 7. Only (Down, Right) = (8,7) is doubly circled, so that is the unique pure-strategy Nash equilibrium.
A cell with only one circled payoff is not a Nash equilibrium. Equilibrium requires mutual best response, so the same cell must contain one circled payoff for each player.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The speaker names the concept as “Nash equilibrium.”
The whiteboard title reads “Nash Equilibrium,” and later a written note appears as “Nash eqbm.”
Nash equilibrium / Nash eqbm
An equilibrium notion described verbally as a “no regrets equilibrium” and written on the board as “Best response to best response.”
Two-player normal-form games represented by payoff matrices.
“Harry’s payoffs in this case are red. He’s player one, so his payoffs are always the first payoffs listed.”
In the Harry-Hermione matrix, the first number in each cell is written in red.
first payoff entry (red)
Player 1’s payoff in the displayed two-player matrix.
Each cell of the Harry-Hermione payoff matrix.
“look at the whole thing from Hermione’s perspective, looking only at the second payoffs listed since she is player two.”
In the Harry-Hermione matrix, the second number in each cell is written in purple.
second payoff entry (purple)
Player 2’s payoff in the displayed two-player matrix.
Each cell of the Harry-Hermione payoff matrix.
“His strategies are up and down.”
Rows of the Harry-Hermione matrix are labeled Up and Down.
Up, Down
Harry’s available pure strategies.
Row choices in the Harry-Hermione game.
The speaker refers to “each of Hermione’s strategies.”
Columns of the Harry-Hermione matrix are labeled Left and Right.
Left, Right
Hermione’s available pure strategies.
Column choices in the Harry-Hermione game.
“Starting from Elizabeth’s perspective, we look at all Elizabeth’s payoffs, and yet we check Mr. Darcy’s strategies.”
The upper-right matrix is headed with “Mr. Darcy” and has row labels Up and Down and column labels Left and Right.
The speaker says “Elizabeth,” but the visible row label in that matrix is not fully legible in the sampled frames; the audio identifies the player as Elizabeth.
The exact color assignment for Elizabeth’s payoffs in the second matrix is not explicitly stated in the clip.
Elizabeth / Mr. Darcy
Names used for the players of the next payoff matrix after the Harry-Hermione example.
Upper-right 2x2 payoff matrix.
Whiteboard title at the top center reads "Nash Equilibrium".
Speaker repeatedly refers to outcomes as Nash equilibria and explains them via best responses.
Nash Equilibrium
A strategy profile in which each player is choosing a best response to the other player's strategy; the speaker also describes it as a "no regrets" strategy.
Two-player normal-form games represented by payoff matrices.
Speaker says Elizabeth is Player 1 and later Emma is Player 1, meaning she cares about the first payoff listed.
In each cell such as (5,9), (4,8), (9,4), and (7,6), the first number is treated as the row player's payoff when circling best responses.
First payoff in each ordered pair
Payoff to Player 1, who is the row player in the examples shown on the board.
All four 2x2 payoff matrices visible on the whiteboard.
Speaker says Mr. Darcy is Player 2 and later Mr. Knightley's perspective is the second payoff.
In cells such as (5,9), (4,8), (9,4), and (7,6), the second number is circled when checking the column player's best response.
Second payoff in each ordered pair
Payoff to Player 2, who is the column player in the examples shown on the board.
All four 2x2 payoff matrices visible on the whiteboard.
Top-right matrix labels the row player as Elizabeth with strategies Up and Down.
Speaker compares Elizabeth's payoffs conditional on Mr. Darcy choosing Left or Right.
Elizabeth: Up / Down
Row-player strategies in the Elizabeth vs. Mr. Darcy example.
Top-right 2x2 matrix.
Top-right matrix labels the column player as Mr. Darcy with strategies Left and Right.
Speaker checks Mr. Darcy's preferred payoff given Elizabeth goes Up or Down.
Mr. Darcy: Left / Right
Column-player strategies in the Elizabeth vs. Mr. Darcy example.
Top-right 2x2 matrix.
Bottom-left matrix labels Ron with strategies Up and Down.
Speaker says Ron is Player 1 and underlines all of the Player 1 strategies before comparing payoffs.
Ron: Up / Down
Row-player strategies in the Ron vs. Neville example.
Bottom-left 2x2 matrix.
“What is Nash equilibrium? Well, it’s basically a no regrets equilibrium where every single player … can say, given what the other player or players did, I’m happy with my choice.”
The video defines Nash equilibrium informally as an outcome in which, after everyone has chosen, each player is satisfied with their own choice given the choices made by the others. This is presented as a verbal characterization rather than a formal mathematical statement.
Applies to strategic interactions among multiple players.
The definition is given for outcomes after all players have chosen strategies.
“Another way of saying that is it’s a best response to a best response, meaning both players can say, given what they did, my best response is the choice I made. And the other player can say the same thing.”
The presenter writes “Nash eqbm.” and then “Best response to best response.” on the whiteboard.
The clip gives a second formulation of Nash equilibrium: each player’s chosen strategy is a best response to the other player’s chosen strategy. The board writing condenses this to “Best response to best response.”
Used for two-player examples in the clip.
Each player’s strategy must be optimal relative to the other player’s fixed strategy.
“Now that doesn’t necessarily mean best choice or best outcome, but let’s actually solve these problems so you can see how it’s done.”
The speaker explicitly warns that a Nash equilibrium is not automatically the best collective result. This prepares the later example where the equilibrium differs from the cell both players would prefer.
Concerns interpretation of equilibrium outcomes in games.
“So we’ve got a game with Harry and Hermione, and Harry’s payoffs in this case are red. He’s player one, so his payoffs are always the first payoffs listed. His strategies are up and down.”
The Harry-Hermione matrix shows rows Up/Down, columns Left/Right, and cells containing ordered pairs such as (4,8), (7,4), (6,3), (9,2).
The video uses a 2x2 table to represent a two-player game. One player chooses rows and the other chooses columns; each cell contains an ordered pair of payoffs, with the first entry belonging to the row player and the second to the column player.
Demonstrated on the Harry-Hermione matrix.
Player 1 corresponds to the first listed payoff; Player 2 to the second.
“So to solve these, we first start by thinking from Harry’s perspective, meaning we only care about the red outcomes. And yet … we check Hermione’s strategies.” The speaker then compares 4 vs 6 and 7 vs 9 for Harry, and later compares 8 vs 4 and 3 vs 2 for Hermione.
The presenter circles the preferred payoff numbers in the Harry-Hermione matrix while comparing entries within a fixed column for Harry and within a fixed row for Hermione.
To find a player’s best response, hold the opponent’s strategy fixed and compare that player’s own payoffs across their available strategies. In the matrix, this means scanning one column at a time for the row player and one row at a time for the column player, then marking the larger personal payoff.
Shown for two-player finite games in normal form.
Requires comparing only the payoff component belonging to the player under consideration.
“So he actually has a dominant strategy, meaning his best response is always down, no matter what she does.”
The clip defines a dominant strategy as a strategy that remains a player’s best response regardless of what the other player does. Harry’s strategy Down is identified as dominant because it beats Up in both of Hermione’s possible columns.
Introduced in the Harry-Hermione example.
Requires comparison across all opponent strategies.
“And then anytime you have two circles in the same box, that means it’s a best response to a best response, or Nash equilibrium.”
In the Harry-Hermione matrix, the cell (6,3) ends up with both payoff entries circled.
After marking each player’s best responses separately, the video uses a visual rule: if both numbers in the same cell are circled, that cell is a Nash equilibrium because each player is simultaneously playing a best response to the other.
Applied to the worked 2x2 matrix example.
Both players’ best-response marks must fall in the same cell.
Bottom text on the whiteboard reads: Nash eqm: "No regrets", Best response to best response.
Speaker concludes that every box with two circles is a best response to a best response or a no regrets strategy.
The video defines a Nash equilibrium operationally as an outcome where both players are playing a best response to each other. The speaker summarizes this as a "no regrets" strategy: once the other player's choice is known, neither player would want to switch.
Applies to the two-player normal-form games shown on the board.
Each player's strategy must be a best response to the other player's strategy.
Throughout the clip the speaker repeatedly says one player checks the other player's strategies, compares payoffs, then flips perspectives.
Best-response marking is done by circling the larger payoff for the relevant player within a fixed row or column.
To find Nash equilibria, fix one player's strategy and circle that opponent's best payoff in the corresponding row or column. Then switch perspectives and do the same for the other player. Any cell containing two circled payoffs, one for each player, is a Nash equilibrium because both choices are mutual best responses.
The game is written as a payoff matrix with ordered pairs.
Player 1's payoff is the first entry and Player 2's payoff is the second entry.
Comparison is made within a fixed row for the column player and within a fixed column for the row player.
Speaker asks what to do if the game is not nice and color-coded, then says she sometimes underlines the payoffs to organize her thoughts.
The earlier examples use colored writing, while the Ron/Neville example uses underlining instead.
When the matrix is not visually color-coded by player, the speaker recommends underlining all of Player 1's payoffs first. This makes it easier to separate whose payoff is being compared when circling best responses.
Useful when the two players' payoffs are not already distinguished by color or another clear visual cue.
“So he actually has a dominant strategy, meaning his best response is always down, no matter what she does.”
In the Harry-Hermione matrix, Harry’s red payoffs are 4 when Up and 6 when Down under Left, and 7 when Up and 9 when Down under Right.
In the Harry-Hermione payoff matrix shown, Down strictly dominates Up for Harry.
Use the displayed payoff matrix with Harry as the row player.
Harry compares only his own first-entry payoffs.
Hermione’s strategies are Left and Right.
For each Hermione strategy (Left and Right), Harry’s payoff from Down exceeds his payoff from Up.
“anytime you have two circles in the same box, that means it’s a best response to a best response, or Nash equilibrium.” Then: “both players would prefer this box up here, but yet this is the Nash equilibrium.”
The cell containing (6,3) is the one with both entries circled in the Harry-Hermione matrix.
For the displayed Harry-Hermione matrix, the Nash equilibrium is the outcome corresponding to Harry choosing Down and Hermione choosing Left, with payoff pair (6,3).
Harry is the row player with payoffs Up/Down.
Hermione is the column player with payoffs Left/Right.
Best responses are determined by maximizing each player’s own payoff entry.
There exists a cell in the matrix where both players’ best responses coincide; that cell is (Down, Left) = (6,3).
“Now we notice here that we have an interesting situation because both players would prefer this box up here, but yet this is the Nash equilibrium.”
The presenter points to another cell while identifying (6,3) as the equilibrium cell.
The exact cell being pointed to as “this box up here” is visually indicated but not named aloud in the clip; from the displayed matrix it corresponds to the cell (7,4).
In the Harry-Hermione example, the equilibrium cell (6,3) is different from the cell that the speaker says both players would prefer.
Use the same displayed Harry-Hermione payoff matrix.
Interpret preference by comparing payoff pairs across cells.
There is at least one non-equilibrium cell that the speaker describes as jointly preferred over the equilibrium cell.
“And this is actually a prisoner’s dilemma. I won’t get into that here, but let’s do this next box.”
The clip states the classification but does not prove it or define prisoner’s dilemma within this segment.
The speaker identifies the worked Harry-Hermione game as an instance of a prisoner’s dilemma.
The identification is based on the displayed payoff structure of the Harry-Hermione game.
The specific example shown is asserted to belong to the class of prisoner’s dilemma games.
Speaker says, "So in this box we actually have two different Nash equilibria."
Top-right matrix shows circled payoffs in cells (Up, Left) = (5,9) and (Down, Right) = (4,7).
In the Elizabeth/Mr. Darcy payoff matrix, the outcomes (Up, Left) and (Down, Right) are both Nash equilibria.
Elizabeth is the row player with strategies Up and Down.
Mr. Darcy is the column player with strategies Left and Right.
Payoffs are read as (Elizabeth, Mr. Darcy).
A Nash equilibrium is a cell where both players' payoffs are circled as best responses.
For the specific 2x2 matrix shown on the board.
Speaker says, "So in this game we have no Nash equilibrium."
Bottom-left matrix has no cell containing two circled payoffs after the best-response analysis.
In the Ron/Neville payoff matrix shown, there is no pure-strategy Nash equilibrium.
Ron is the row player with strategies Up and Down.
Neville is the column player with strategies Left and Right.
Payoffs are read as (Ron, Neville).
Nash equilibrium is identified by a cell with mutual best responses.
For the specific 2x2 matrix shown on the board.
Bottom-right matrix ends with both payoffs circled only in the cell (Down, Right) = (8,7).
Speaker generalizes that every box with two circles is a best response to a best response.
The speaker does not explicitly say the words "one Nash equilibrium" for this final matrix, but the visual marking identifies exactly one mutual-best-response cell.
In the Emma/Mr. Knightley payoff matrix shown, the unique pure-strategy Nash equilibrium is (Down, Right) with payoffs (8,7).
Emma is the row player with strategies Up and Down.
Mr. Knightley is the column player with strategies Left and Right.
Payoffs are read as (Emma, Mr. Knightley).
A Nash equilibrium is a cell where both players' payoffs are circled as best responses.
For the specific 2x2 matrix shown on the board.
The speaker narrates the full solution: compare Harry’s payoffs column by column, then Hermione’s payoffs row by row, then use the two-circle rule.
Circles are drawn around 6, then 9, then 8, then 3 in the Harry-Hermione matrix.
Start from the displayed 2x2 payoff matrix with Harry as row player and Hermione as column player.
Observed directly on the whiteboard.
Hold Hermione’s strategy fixed and inspect Harry’s own payoffs in that column.
Best-response procedure explained in the audio.
Choose the larger red payoff in the Left column.
Definition of best response by payoff maximization.
Repeat the same comparison in the Right column.
Best-response procedure explained in the audio.
Choose the larger red payoff in the Right column.
Definition of best response by payoff maximization.
Since Down wins in both columns, it is optimal regardless of Hermione’s choice.
This is exactly the speaker’s stated conclusion that Harry has a dominant strategy.
Switch perspectives and hold Harry’s row fixed while comparing Hermione’s second-entry payoffs.
Audio: “looking only at the second payoffs listed since she is player two.”
Choose the larger purple payoff in the Up row.
Best-response comparison for the column player.
Repeat the comparison in the Down row.
Audio explicitly states this comparison.
Choose the larger purple payoff in the Down row.
Best-response comparison for the column player.
Marking Harry’s best response in the Left column gives 6; marking Hermione’s best response in the Down row gives 3; both marks land in the same cell.
Visual evidence from the board plus the two-circle rule stated aloud.
Because both players are simultaneously playing best responses to each other in that cell.
“anytime you have two circles in the same box, that means it’s a best response to a best response, or Nash equilibrium.”
The worked matrix has Nash equilibrium (Down, Left) with payoff pair (6,3).
“both players would prefer this box up here, but yet this is the Nash equilibrium. And this is actually a prisoner’s dilemma.”
The equilibrium cell is (6,3), while another cell in the matrix is pointed out as jointly preferred.
The clip does not spell out the formal Pareto-comparison argument; it only asserts the preference relation verbally while pointing.
From the previous derivation, the two-circle cell is (6,3).
Derived from best-response markings.
The video contrasts the equilibrium with a different outcome that is described as better for both players.
Direct audio statement plus pointing gesture.
The example illustrates that mutual best response does not guarantee the most preferred joint result.
Matches the earlier warning that equilibrium “doesn’t necessarily mean best choice or best outcome.”
The Harry-Hermione game demonstrates that a Nash equilibrium may be worse for both players than some other outcome, which the speaker labels as a prisoner’s dilemma.
Speaker compares Elizabeth's payoffs conditional on Darcy's move, then flips to Darcy's payoffs conditional on Elizabeth's move.
She circles 5 and 4 for Elizabeth, then 9 and 7 for Mr. Darcy in the top-right matrix.
Start from the top-right payoff matrix with Elizabeth as row player and Mr. Darcy as column player.
Observed directly on the whiteboard.
Fix Mr. Darcy's columns and choose Elizabeth's better payoff in each column: if Darcy plays Left, Elizabeth prefers 5 to 3; if Darcy plays Right, she prefers 4 to 2.
Best response definition applied to Player 1's payoffs within each column.
Circle 5 in (Up, Left) and 4 in (Down, Right) as Elizabeth's best responses.
Visual marking follows the numerical comparison just made.
Now fix Elizabeth's rows and choose Mr. Darcy's better payoff in each row: if Elizabeth plays Up, Darcy prefers 9 to 6; if she plays Down, he prefers 7 to 5.
Best response definition applied to Player 2's payoffs within each row.
Circle 9 in (Up, Left) and 7 in (Down, Right) as Mr. Darcy's best responses.
Visual marking follows the row-wise comparison.
The cells with two circled payoffs are exactly (Up, Left) and (Down, Right), so these are the Nash equilibria.
By the video's rule that a box with two circles is a best response to a best response.
The Elizabeth vs. Mr. Darcy game has two pure-strategy Nash equilibria: (Up, Left) and (Down, Right).
Speaker introduces the underline tip, then compares Ron's payoffs by column and Neville's by row, ending with "we have no Nash equilibrium."
In the bottom-left matrix she underlines first payoffs, circles 9 and 7 for Ron, then circles 8 and 6 for Neville.
Begin with the bottom-left matrix, where Ron is Player 1 and Neville is Player 2.
Observed directly on the whiteboard.
Underline all of Player 1's payoffs to keep track of Ron's values in an uncolored matrix.
Speaker's stated organizational method.
For each Neville column, compare Ron's payoffs: if Neville plays Left, Ron prefers 9 to 5; if Neville plays Right, Ron prefers 7 to 2.
Best response definition for the row player within each column.
Mark Ron's best responses by circling those underlined payoffs.
Visual implementation of the previous comparisons.
Switch to Neville's perspective and compare the second payoff within each Ron row: if Ron plays Up, Neville prefers 8 to 4; if Ron plays Down, Neville prefers 6 to 3.
Best response definition for the column player within each row.
Mark Neville's best responses in the remaining cells.
Visual implementation of the row-wise comparisons.
After both players' best responses are marked, no single outcome has mutual best responses.
Direct inspection of the completed matrix.
The Ron vs. Neville game has no pure-strategy Nash equilibrium.
Speaker says Emma is Player 1, compares first payoffs by column, then flips to Mr. Knightley's second payoffs by row.
In the bottom-right matrix she circles 7 and 8 for Emma, then 4 and 7 for Mr. Knightley, leaving only (Down, Right) doubly circled.
Start from the bottom-right matrix with Emma as row player and Mr. Knightley as column player.
Observed directly on the whiteboard.
For each Mr. Knightley column, compare Emma's payoffs: if he plays Left, Emma prefers 7 to 4; if he plays Right, Emma prefers 8 to 5.
Best response definition for Player 1 within each column.
Mark Emma's best responses in the down row cells.
Visual marking follows the column-wise comparison.
Now compare Mr. Knightley's payoffs within each Emma row: if Emma plays Up, he prefers 4 to 3; if Emma plays Down, he prefers 7 to 6.
Best response definition for Player 2 within each row.
Mark Mr. Knightley's best responses in the right-column cells.
Visual marking follows the row-wise comparison.
Inspect the completed matrix: the sole cell containing both players' circled best responses is (Down, Right).
By the video's criterion that two circles in one box indicate mutual best response.
The Emma vs. Mr. Knightley game has one pure-strategy Nash equilibrium: (Down, Right) with payoffs (8,7).
The entire middle portion solves the Harry-Hermione game step by step.
A 2x2 matrix labeled with Harry/Hermione and entries (4,8), (7,4), (6,3), (9,2) is annotated live.
Find the Nash equilibrium of the displayed two-player game between Harry and Hermione.
Harry is player 1 and chooses rows Up or Down.
Hermione is player 2 and chooses columns Left or Right.
Payoff matrix entries are (Up, Left)=(4,8), (Up, Right)=(7,4), (Down, Left)=(6,3), (Down, Right)=(9,2).
Identify each player’s best responses and determine the Nash equilibrium cell.
Fix Hermione at Left and see which row gives Harry more.
Best-response method stated in the clip.
Circle the larger first-entry payoff.
Payoff maximization for player 1.
Fix Hermione at Right and repeat the comparison.
Same best-response method.
Thus Down is always Harry’s best reply.
Speaker explicitly calls this a dominant strategy.
Fix Harry at Up and inspect the second entries.
Perspective switch explained in the audio.
Circle the larger second-entry payoff in that row.
Payoff maximization for player 2.
Fix Harry at Down and repeat.
Same method for the other row.
Thus Left is Hermione’s best reply to either Harry action.
Directly follows from the two comparisons.
Both players’ best-response marks fall in the same cell, so that cell is the equilibrium.
Two-circle rule stated aloud.
The Nash equilibrium is (Down, Left) with payoff pair (6,3).
The result is checked by confirming that 6 is Harry’s largest payoff in the Left column and 3 is Hermione’s largest payoff in the Down row, so neither player wants to deviate unilaterally.
“but let’s do this next box. Starting from Elizabeth’s perspective, we look at all Elizabeth’s payoffs, and yet we check Mr. Darcy’s strategies.”
The presenter moves to the upper-right matrix headed “Mr. Darcy,” with visible entries (5,9), (2,6), (3,5), (4,7).
The clip ends before the second example is solved.
The speaker says “Elizabeth,” while the visible row label in the matrix is not fully legible in the sampled frames.
It is not shown which payoff component belongs to Elizabeth versus Mr. Darcy in this second matrix.
Begin solving the next 2x2 payoff matrix after the Harry-Hermione example.
The next matrix is the upper-right one on the board.
The speaker refers to Elizabeth and Mr. Darcy.
Visible entries include (5,9), (2,6), (3,5), (4,7).
Start the same best-response procedure from Elizabeth’s perspective.
The presenter transitions from the solved Harry-Hermione game to the upper-right example.
Audio cue “let’s do this next box.”
This mirrors the earlier method: fix the opponent’s strategy and compare only one player’s own payoffs.
Explicitly stated in the final seconds.
No final equilibrium is computed within this clip; only the setup for the next example is given.
Cannot verify a final answer because the solution process stops before completion in the provided segment.
Top-right matrix labeled Elizabeth and Mr. Darcy with entries (5,9), (2,6), (3,5), (4,7).
Speaker works through both players' preferences and states there are two Nash equilibria.
Find the Nash equilibria of the 2x2 game between Elizabeth and Mr. Darcy.
Elizabeth is Player 1 with strategies Up and Down.
Mr. Darcy is Player 2 with strategies Left and Right.
Payoff matrix: (Up, Left)=(5,9), (Up, Right)=(2,6), (Down, Left)=(3,5), (Down, Right)=(4,7).
Identify all pure-strategy Nash equilibria.
If Mr. Darcy chooses Left, Elizabeth compares 5 and 3 and prefers 5.
Best response for Player 1 in the Left column.
If Mr. Darcy chooses Right, Elizabeth compares 4 and 2 and prefers 4.
Best response for Player 1 in the Right column.
If Elizabeth chooses Up, Mr. Darcy compares 9 and 6 and prefers 9.
Best response for Player 2 in the Up row.
If Elizabeth chooses Down, Mr. Darcy compares 7 and 5 and prefers 7.
Best response for Player 2 in the Down row.
The cells with mutual best responses are (Up, Left) and (Down, Right).
Both corresponding payoffs are circled in those cells.
Two Nash equilibria: (Up, Left) and (Down, Right).
Each listed cell contains one circled payoff for Elizabeth and one circled payoff for Mr. Darcy, matching the video's "two circles in the same box" test.
Bottom-left matrix labeled Ron and Neville with entries (9,4), (2,8), (5,6), (7,3).
Speaker demonstrates the underline method and concludes there is no Nash equilibrium.
Find the Nash equilibria of the 2x2 game between Ron and Neville when the matrix is not color-coded.
Ron is Player 1 with strategies Up and Down.
Neville is Player 2 with strategies Left and Right.
Payoff matrix: (Up, Left)=(9,4), (Up, Right)=(2,8), (Down, Left)=(5,6), (Down, Right)=(7,3).
Identify all pure-strategy Nash equilibria.
Underline all first-payoff entries to identify Ron's payoffs.
Speaker's recommended organization method for uncolored matrices.
Ron's best responses are Up against Left and Down against Right.
Compare Ron's payoffs within each column.
Neville's best responses are Right against Up and Left against Down.
Compare Neville's payoffs within each row.
No outcome is a best response for both players simultaneously.
Inspection of the completed best-response markings.
No pure-strategy Nash equilibrium.
The video's criterion requires two circles in the same box; none appears in the finished Ron/Neville matrix.
Bottom-right matrix labeled Emma and Mr. Knightley with entries (4,3), (5,4), (7,6), (8,7).
Speaker says Emma is Player 1, works through both perspectives, and points to the doubly circled cell.
Find the Nash equilibrium of the 2x2 game between Emma and Mr. Knightley.
Emma is Player 1 with strategies Up and Down.
Mr. Knightley is Player 2 with strategies Left and Right.
Payoff matrix: (Up, Left)=(4,3), (Up, Right)=(5,4), (Down, Left)=(7,6), (Down, Right)=(8,7).
Identify the pure-strategy Nash equilibrium.
Emma's best response is Down whether Mr. Knightley plays Left or Right.
Compare Emma's first payoffs within each column.
Mr. Knightley's best response is Right whether Emma plays Up or Down.
Compare his second payoffs within each row.
Only the bottom-right cell receives both players' circled best responses.
Mutual-best-response test from the video.
One Nash equilibrium: (Down, Right) with payoffs (8,7).
The bottom-right cell is the only box with two circles, matching the speaker's final rule.
The whiteboard shows the title “Nash Equilibrium” and four 2x2 payoff matrices arranged in a grid.
Title “Nash Equilibrium”
Four 2x2 payoff matrices
Presenter holding markers
The presenter faces the camera and gestures toward the board while introducing the topic.
The board already contains multiple example matrices before any new writing begins.
The visual setup indicates that the lesson will explain Nash equilibrium through several worked 2x2 games.
The presenter writes “Nash eqbm.” and then “Best response to best response.” beneath the matrices.
Handwritten text “Nash eqbm.”
Handwritten text “Best response to best response.”
New text is added below the matrices as the speaker restates the definition.
The four example matrices remain visible above the new writing.
The board text converts the spoken informal definition into a short mnemonic phrase.
Red circles are drawn around 6 and then 9 in the Harry-Hermione matrix while the speaker compares Harry’s payoffs column by column.
Harry-Hermione matrix
Red circles around 6 and 9
First the 6 in (6,3) is circled, then the 9 in (9,2) is circled.
Only Harry’s first-entry payoffs are being marked in this stage.
The visual marks encode Harry’s best responses to Hermione’s Left and Right choices.
Purple circles are drawn around 8 and then 3 in the Harry-Hermione matrix while the speaker switches to Hermione’s perspective.
Harry-Hermione matrix
Purple circles around 8 and 3
First the 8 in (4,8) is circled, then the 3 in (6,3) is circled.
Only Hermione’s second-entry payoffs are being marked in this stage.
These marks encode Hermione’s best responses to Harry’s Up and Down choices.
The cell (6,3) contains both a red circle and a purple circle; the presenter points to it and then to another cell while discussing preference.
The exact second cell pointed to is indicated visually but not named aloud in the clip.
Cell (6,3) with two circles
Presenter’s pointing hand
Attention shifts to the doubly marked cell, then to another cell described as jointly preferred.
The matrix entries themselves do not change during this explanation.
The overlapping circles visually demonstrate the “best response to best response” criterion for Nash equilibrium.
The whiteboard shows the title "Nash Equilibrium" at top center and four 2x2 payoff matrices arranged in a grid, plus a summary line at the bottom.
Some left-side matrices are partially obscured by the presenter at certain moments.
Title "Nash Equilibrium"
Top-right Elizabeth vs. Mr. Darcy matrix
Bottom-left Ron vs. Neville matrix
Bottom-right Emma vs. Mr. Knightley matrix
Additional partially visible matrices on the left
Bottom summary text
Presenter moves among the matrices and adds circles or underlines as she explains each example.
The board remains organized as multiple 2x2 normal-form games with row strategies Up/Down and column strategies Left/Right.
The bottom summary text stays visible as the definitional anchor.
The visual layout supports comparing several small games side-by-side using the same best-response marking method.
In the top-right matrix, the presenter circles 5 and 4 for Elizabeth, then circles 9 and 7 for Mr. Darcy.
Matrix cells (5,9), (2,6), (3,5), (4,7)
Red circles around selected payoff numbers
First Elizabeth's best payoffs are circled column by column.
Then Mr. Darcy's best payoffs are circled row by row.
Two cells end up with both payoffs circled.
The underlying payoff numbers do not change.
Player roles remain Elizabeth as row player and Mr. Darcy as column player.
The animation visually implements the mutual-best-response test and reveals two Nash equilibria.
In the bottom-left matrix, the presenter underlines the first payoff in each cell and then circles best responses for Ron and Neville.
Matrix cells (9,4), (2,8), (5,6), (7,3)
Underlines beneath 9, 2, 5, 7
Circles around selected best responses
Underlining is added first to distinguish Player 1's payoffs.
Ron's best responses are circled in the underlined entries.
Neville's best responses are circled in the second entries.
No cell ends with two circles.
The matrix values themselves remain unchanged.
The row/column strategy labels remain Up/Down and Left/Right.
This sequence demonstrates a practical notation trick for uncolored matrices and shows absence of a pure Nash equilibrium.
In the bottom-right matrix, the presenter circles Emma's best responses 7 and 8, then Mr. Knightley's best responses 4 and 7, leaving only (8,7) doubly circled.
Matrix cells (4,3), (5,4), (7,6), (8,7)
Circles around 7, 8, 4, and 7
Emma's column-wise best responses are marked first.
Mr. Knightley's row-wise best responses are marked second.
The bottom-right cell becomes the unique doubly circled outcome.
The payoff ordering remains (Emma, Mr. Knightley).
The matrix structure remains 2x2 with Up/Down versus Left/Right.
The visual result directly exhibits the unique pure-strategy Nash equilibrium.
“Now that doesn’t necessarily mean best choice or best outcome…”
A Nash equilibrium is the best possible result for the players taken together.
The video explicitly warns that equilibrium does not necessarily mean the best outcome, and the later Harry-Hermione example shows an equilibrium cell different from the cell both players would prefer.
“it’s a best response to a best response, meaning both players can say, given what they did, my best response is the choice I made.”
Calling a move a best response means it is the globally best move independent of others’ actions.
In the clip, best response is always relative to the other player’s chosen strategy; the comparison is made after fixing the opponent’s action.
Speaker explicitly warns about games that are not color-coded and recommends underlining Player 1's payoffs to stay organized.
In an uncolored payoff matrix, one may accidentally compare the wrong entries and treat Player 2's payoff as if it were Player 1's.
The video's remedy is to first mark Player 1's payoffs consistently, for example by underlining them, before circling best responses.
Speaker says that a box with two circles is a best response to a best response.
Cells with only one circled payoff are not identified as equilibria.
A cell with only one player's best response circled may be mistaken for a Nash equilibrium.
A Nash equilibrium requires mutual best response, so the same cell must contain two circles, one for each player.
The speaker says “Another way of saying that is it’s a best response to a best response.”
The clip presents the no-regrets description and the best-response-to-best-response description as two equivalent ways of stating the same idea.
“anytime you have two circles in the same box, that means it’s a best response to a best response, or Nash equilibrium.”
The two-circle visual rule is the matrix-based application of the abstract “best response to best response” definition.
The speaker first explains the matrix layout and then uses it to carry out the best-response comparisons.
Understanding which payoff entry belongs to which player is required before applying the column-by-column and row-by-row best-response method.
Harry’s Down is called dominant, and this fact is used while solving the matrix.
The concept of a dominant strategy is instantiated in the worked example by Harry always preferring Down.
After noting that both players would prefer another cell, the speaker says, “And this is actually a prisoner’s dilemma.”
The clip does not define prisoner’s dilemma or prove the classification.
The worked Harry-Hermione game is presented as a specific instance of a prisoner’s dilemma because its equilibrium is not the jointly preferred outcome.
The whole clip uses circling best responses to arrive at the final definition of Nash equilibrium.
Bottom text equates Nash equilibrium with "Best response to best response."
The best-response circling method is the operational procedure by which the video identifies outcomes satisfying the definition of Nash equilibrium.
Speaker introduces underlining specifically as a way to help organize the best-response search when colors are absent.
Underlining Player 1's payoffs is a supporting technique that makes the best-response method less error-prone.
Two doubly circled cells appear in the Elizabeth/Darcy matrix.
Speaker concludes there are two Nash equilibria.
The claim that the Elizabeth/Darcy game has two equilibria depends on applying the mutual-best-response marking procedure.
No cell in the Ron/Neville matrix receives two circles.
Speaker concludes there is no Nash equilibrium.
The absence of a pure Nash equilibrium is established by showing that no outcome is a mutual best response.
Exactly one cell, (Down, Right), is doubly circled in the Emma/Knightley matrix.
Speaker uses the two-circle rule to identify best-response-to-best-response outcomes.
The uniqueness of the equilibrium in the Emma/Knightley example follows from the same mutual-best-response test.
Opening definition and restatement of Nash equilibrium.
“it’s a best response to a best response”
Board text “Best response to best response.”
Explanation that Harry is player one and his payoffs are the first listed.
2x2 matrix with ordered pairs.
Step-by-step comparison of 4 vs 6, 7 vs 9, 8 vs 4, and 3 vs 2.
Circles placed on best-response payoffs.
“he actually has a dominant strategy, meaning his best response is always down”
Warning that equilibrium is not necessarily best outcome, followed by the jointly preferred cell remark.
“And this is actually a prisoner’s dilemma.”
The clip does not elaborate on the definition.
Bottom text reads: Nash eqm: "No regrets", Best response to best response.
Speaker defines equilibrium through mutual best response.
Speaker repeatedly says to check one player's strategies, compare payoffs, then flip perspectives.
Best responses are found by circling larger payoffs row-wise and column-wise.
Top-right matrix ends with two doubly circled cells.
Speaker says there are two different Nash equilibria in this box.
Bottom-left matrix has no cell with two circles after marking best responses.
Speaker says, "So in this game we have no Nash equilibrium."
Bottom-right matrix leaves only (Down, Right) = (8,7) doubly circled.
Speaker identifies the doubly circled box as a best response to a best response.
Covered · Spoken introduction defining Nash equilibrium as a no-regrets outcome.
Covered · Restatement of the definition and writing of “Best response to best response.” on the board.
Covered · Warning that equilibrium need not be the best overall outcome.
Covered · Introduction of the Harry-Hermione matrix and explanation of payoff ordering.
Covered · Harry’s best responses are found by comparing payoffs column by column.
Covered · Conclusion that Harry has a dominant strategy, Down.
Covered · Perspective switch to Hermione and comparison of her payoffs row by row.
Covered · Rule that a cell with two circles is a Nash equilibrium; identification of (6,3).
Covered · Discussion that both players would prefer another cell and labeling the example as a prisoner’s dilemma.
Covered · Transition to the next matrix with Elizabeth and Mr. Darcy; only setup is shown before the clip ends.
Covered · Top-right Elizabeth vs. Mr. Darcy example is fully worked and yields two Nash equilibria.
Covered · Speaker transitions from color-coded examples to a tip for uncolored matrices.
Covered · Bottom-left Ron vs. Neville example demonstrates underlining and shows no pure Nash equilibrium.
Covered · Bottom-right Emma vs. Mr. Knightley example produces one equilibrium and the clip ends with the verbal definition of Nash equilibrium.
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