The Nash equilibrium of a game is the strategy profile s∗=(si∗,s−i∗) which illustrates all players in a game using their best response, such that for all strategies si∈Si of all players i∈N, ui(si∗,s−i∗)≥ui(si,s−i∗).
The payoff that occurs when each player adheres to s∗ is the Nash equilibrium outcome.
Intuition
In a Nash equilibrium outcome, all players know (and expect) the strategy of every other player in the game. In a way, a Nash equilibrium is a stable prescription for play where a single player does not have a positive unilateral incentive to deviate from the equilibrium, although they might have an indifferent incentive (since we use greater than or equal to '≥' instead of strictly greater than '>'). By deviating from the Nash equilibrium, a single player would simply lower their own payoffs if they think the other players will not deviate from their respective strategies, and this is inconsistent with rational play.
Something else to consider is the Nash equilibrium as a focal point for players. Since a Nash equilibrium utilizes each player’s most rational response, a player could ask themselves what they expect as the outcome of the game by accounting for unreasonable plays (mistakes) by other players. If they expect everyone to play rationally (or coordinate on a Nash equilibrium by pre-play communication), then they are expecting a Nash equilibrium. If mistakes do happen from players, they could be corrected by changes in strategy with trial and error until a Nash equilibrium is reached.
Finally, we can find a Nash equilibrium by identifying the best response si∗ of player iS−i of −i. Then, if i‘s best response to −i is the best response of −i to i, this is a Nash equilibrium. Note that there may be more than one Nash equilibria in a game.
Essentially, when looking for Nash equilibria, ask yourself whether any of the players i∈N have an incentive to deviate from the current strategy profile (change their choice).
Properties
(Nash’s Theorem) In a finite strategic form game, there is always at least one Nash equilibrium.
(Dominance Subsumption) Any game with a dominant strategy also has a unique Nash equilibrium.
Pre-last video of week 1 + starting from pg 80 from chapter 5 of textbook
(Rule of Thumb) A finite strategic form game has an odd number of total NEs and at most one mixed strategy Nash equilibrium.
A more elaborate table of this can be seen below:
Pure NEs
Mixed NEs
2
1
1
0
0
1
Examples
The Contributions Game, Revisited
Recall the contributions game from before, in which two players can choose to contribute or to free-ride (not contribute) on a public good. The game matrix is as follows:
\begin{document}\begin{tikzpicture} % Draw the vertical lines \draw (0, 0) -- (0, -3); \draw (1.5, 0) -- (1.5, -3); \draw (3, 0) -- (3, -3); % Draw the horizontal lines \draw (0, 0) -- (3, 0); \draw (0, -1.5) -- (3, -1.5); \draw (0, -3) -- (3, -3); % Add labels for players \node at (1.5, 1) {Player 2}; \node[rotate=90] at (-1, -1.5) {Player 1}; % Add labels for Player 2's choices \node at (0.75, 0.25) {C}; \node at (2.25, 0.25) {N}; % Add labels for Player 1's choices \node at (-0.25, -0.75) {C}; \node at (-0.25, -2.25) {N}; % Add payoffs for each cell \node at (0.75, -0.75) {(1-c, 1-c)}; \node at (2.25, -0.75) {(1-c, 1)}; \node at (0.75, -2.25) {(1, 1-c)}; \node at (2.25, -2.25) {(0, 0)};\end{tikzpicture}\end{document}
In this game there are two Nash equilibria: \begin{equation}(C, N)\end{equation} and \begin{equation}(N, C)\end{equation}, with respective Nash equilibrium outcomes \begin{equation}(1-c, 1)\end{equation} and \begin{equation}(1, 1-c)\end{equation}. These strategy profiles illustrate the best (most rational) responses of each player to the other player.
Nash Equilibrium in the Animal Kingdom
Consider the desert spider species Agelenopsis aperta found in New Mexico; the female lays its eggs within webs, which are scarce since they are difficult to build. As such, females threaten each others for existing webs, although rarely coming to a conflict. Usually, the winner of the conflict is often “differently endowed” from the loser in vital characteristics, and when the stakes are higher, a fight is more likely.
Suppose the utility of the web is 10. We can model this into a game of two players (the two conflicting females), and two strategies: fight or concede. If one of them fights and the other concedes, the one that fights gets the web. If neither fights or both fight, there is a 50-50 chance that either of them get the web, although it they do fight they have the chance to be physically harmed. Let the payoff x denote the payoff of both spiders fighting. If the physical costs are higher than the expected value of the web, x<0; otherwise, x>0:
\begin{document}\begin{tikzpicture} % Draw the vertical lines \draw (0, 0) -- (0, -3); \draw (1.5, 0) -- (1.5, -3); \draw (3, 0) -- (3, -3); % Draw the horizontal lines \draw (0, 0) -- (3, 0); \draw (0, -1.5) -- (3, -1.5); \draw (0, -3) -- (3, -3); % Add labels for players \node at (1.5, 1) {Spider 2}; \node[rotate=90] at (-1, -1.5) {Spider 1}; % Add labels for Player 2's choices \node at (0.75, 0.25) {C}; \node at (2.25, 0.25) {F}; % Add labels for Player 1's choices \node at (-0.25, -0.75) {C}; \node at (-0.25, -2.25) {F}; % Add payoffs for each cell \node at (0.75, -0.75) {(5, 5)}; \node at (2.25, -0.75) {(0, 10)}; \node at (0.75, -2.25) {(10, 0)}; \node at (2.25, -2.25) {(x, x)};\end{tikzpicture}\end{document}
How would we find the Nash equilibrium? Suppose x<0 i. e. the physical costs of a fight are greater than the web’s value. Then, there are two Nash equilibria at (F,C) and (C,F). However, if x>0, then the Nash equilibrium is at (F,F); higher stakes engender more fights.
If the spiders pick (C,C), then weight first and incumbency second matter in who gets the web; when the spiders weigh about the same, the one who already owns the web will keep it. If weight of the two spiders is substantially different, and one suffers less from a fight than the other, then payoff of (F,F) is (x,y) with x<0<y, and the only Nash equilibrium is (C,F).