In one-shot interaction, a strategy si∗ is the best response of player i∈N to a certain strategy combination selected by the other players s−i if for all other strategies si∈Si that player i could use, ui(si∗,s−i)≥ui(si,s−i).
Example
The Contributions Game
Consider a game of players, who have the choice of either contributing (or not) to a public good. Suppose now that if at least one player contributes, all players get 1 unit of utility. However, contributing has a cost of 0<c<1 utility units. We can model this game into a game matrix:
\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}
Let’s first look player 1’s best responses:
Player 1’s best response to player 2 contributing is to not contribute (free-ride), since it is the choice that maximizes player 1’s utility given player 2’s choice. This leads to the payoff (1,1−c).
Similarly, player 1’s best response to player 2 not contributing is to contribute, leading to the payoff (1−c,1).
Now, let’s look at player 2’s best responses:
Player 2’s best response to player 1 contributing is to not contribute, paying off (1−c,1).
Player 2’s best response to player 1 not contributing is to contribute, paying off (1,1−c).
Since there is no dominant strategy, each player should form expectations of how the other player will play.