Nash equilibria: beautiful but problematic
The pay-offs to the two players in this simple, non-zero-sum game are indicated for each possible combination of strategy choices. The rows designate Rose's choice of strategy (A or B) while the columns designate Colin's choice of strategy (A or B). The entry (a,b) indicates that Rose receives pay-off a while Colin receives b. In real life this game might correspond to a situation in which two enemies are deciding whether to live in city A or city B. The pay-off is happiness - each player wants to live in the nicer city and without their enemy nearby. Colin and Rose both prefer city B. However, city B is smaller than city A, and it is therefore easier for them to run into each other. On the other hand, city A is bigger and so it is rather unlikely that they would run into each other - yet the city itself is less appealing. We can model the scenario as shown in the pay-off table, where greater happiness is indicated by larger values. The players have perfect knowledge of the pay-off table, and can be thought of as making their moves at the same time. If Colin chooses A, Rose will gain 1 with strategy A and 5 with strategy B, so she cannot profitably deviate from strategy B (as indicated by the downward arrow in column A). If Colin chooses B, Rose will gain 2 with strategy A but only -1 with strategy B, so she cannot profitably deviate from strategy A (as indicated by the upward arrow in column B). Colin will also reason this way: if Rose chooses A, he should choose B (as indicated by the horizontal arrow to the right in row A). If Rose chooses B, he should choose A (as indicated by the horizontal arrow to the left in row B). A Nash equilibrium is the set of strategy choices that provides the optimal pay-off for each player individually. Therefore, both (A,B) and (B,A) represent Nash equilibria (purple boxes) where the first entry denotes Rose's strategy and the second Colin's. Rose prefers (B,A) while Colin prefers (A,B). How can they choose between them without prior communication? If Rose tries for (B,A) and thus chooses strategy B, while Colin tries for (A,B) and also chooses strategy B, they will end up with the inferior pay-off (-1,-1).