Two players each choose once, without seeing the other's choice, and the table says what each receives. Mark every player's best reply to every move of the other, and the cells where both marks fall are the cells no one wants to leave. The wrong model is to pick the row with the biggest number in it.
Core ideas
Nothing in this course is harder than reading one small table correctly. The three ideas below are the whole of the reading.
Each player has a short list of strategies, and every pair of choices, one from each list, is a cell. The cell holds two numbers: what the row player gets, then what the column player gets. The players choose without seeing each other's choice, so the table is all there is.
Hold the other player's strategy fixed and compare your own payoffs along that line of the table. The largest is your best response to that strategy. Change what the other does and the best response may change with it.
Mark each player's best replies. A cell carrying both players' marks is one where each is already doing the best available against what the other is doing. The next lessons call such a cell an equilibrium.
The material
A table with two numbers in every cell, and a procedure for marking it.
A game in strategic form lists the players, the strategies open to each, and a payoff for each player at every combination of strategies. With two players it is a bimatrix: the row player's strategies run down the side, the column player's run across the top, and each cell shows row payoff, column payoff in that order.
A payoff is a number that stands for how much the outcome is worth to the player who gets it. Where the number comes from is a question the lab cannot answer; it uses whatever you type, and every verdict in this course is a verdict about the payoffs as stated. Decision and Rationality is where the numbers earn their meaning as utilities.
Here is the smallest interesting case. Two people must each choose the left path or the right path to a meeting point, and they cannot talk. If both take the left path they meet at a place both like very much. If both take the right they meet at a place they like less. If they choose differently they do not meet.
A strategy is a best response to a given strategy of the other player when no other strategy of one's own pays more against it. Ties are allowed: a best response need only be among the largest.
To find the row player's best responses, go column by column. If the column player takes L, the row player compares 2 for L with 0 for R, and L is the best response. If the column player takes R, the comparison is 0 for L against 1 for R, and R is the best response. The column player's marks are found the same way, going row by row.
The two cells where both players' marks fall are the L, L cell and the R, R cell. The other two cells carry no pair of marks: in the L, R cell the row player gets 0 and would prefer to be playing R.
A tempting shortcut is to look for the row with the largest number anywhere in it and play that. In the table above the largest number is 2, in row L. But the row player's task is not to find a good row in the abstract; it is to answer what the other player does. Against column R, row L pays 0 and row R pays 1, so the row with the biggest number is the worse reply.
The lab also holds a game whose only marked cell is D, D (each player's best reply is D whatever the other does) and a game, matching pennies, in which no cell carries both marks. In matching pennies one player wins if the two coins match and the other wins if they do not, so in every cell exactly one player would rather have chosen otherwise.
Marking a table is mechanical, and it is meant to be. What is not mechanical is deciding which table describes the situation. The same two people with different feelings about the right-hand meeting place are playing a different game.
Interactive
best responses, equilibria, dominance and efficiency, computed from the payoffs
Method
Five moves, and the third is the one people skip.
Row player's down the side, column player's across the top. Each cell holds two numbers, row first.
Take one column. The row player cannot change what the other has chosen, only answer it.
Not the whole table, and not the other player's numbers. The larger one gets the mark; a tie marks both.
The column player compares along a row, the mirror image of the row player's comparison.
Those are the cells where each player is already best-responding. A table can have one, several, or none.
Payoffs are row, column. Row L, column L gives each of them 2.
The two diagonal cells are each a best response to the other. The off-diagonal cells are not: in each of them one player gets 0 and would switch. Which diagonal cell the pair ends up in is not settled by the table, which is the question “Coordination, Conventions and the Stag Hunt” takes up.
Given any two-by-two bimatrix, mark each player's best replies to each move of the other, list the cells with both marks, and say for one unmarked cell which player would switch and to what.
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