There are games with no Nash equilibrium at all. For such games, we will make predictions about players’ behavior by enlarging the set of strategies to include the possibility of randomization; once players can behave randomly, one of john Nash’s main result establishes that equilibria always exist (Easley and Kleiberg, 2010, p.173).
To prove this statement, we can look at one specific game called matching pennies to explain mixed strategies and mixed strategy equilibrium.
Matching pennies is an example of “attack-defense” games. In such games, one player behaves as the attacker, while the other behaves as the defender. The attacker has two strategies it can choose from – attack A and attack B, while the defender’s strategies are – defend against A or defend against B. If the defender defends against the right attack, the defenders then gets the higher payoff, but if the defenders defends against the wrong attack, the attacker gets the higher payoff (Easley and Kleiberg, 2010, p. 174).
As you see at the link below, matching pennies is a game base in which two players hold a penny each, and simultaneously choose between heads (H) or tails (T). If player 1 and player 2 matches pennies, then one gets the others penny. If the pennies don’t match, one loses its penny to the other.
This is an example zero-sum games, where the payoff of the players sums to zero in every outcome.
A mixed strategy is when a player chooses to randomize over the set of available actions (Sethi, 2007, p.290). Mixed strategies involve a player randomly choosing among pure strategies according to given probabilities (Garrett and Moore, 2008, p. 79).
Since a game consist a set of players, strategies, and payoff, we should notice that by allowing randomization, we have actually changed the game. It no longer consists of two strategies, but a set of strategies corresponding to the interval of number between 0 and 1 (Easley and Kleiberg, 2010, p. 175).
According to John Nash (Nash’s Theorem), there must be at least one Nash equilibrium for all finite games. But in this game, there are no equilibria in pure strategies. The good news is that we have another type of equilibrium called mixed strategy nash equilibrium (Sethi, 2007, p.290).
Mixed Strategy Nash Equilibrium
If no equilibrium exists in pure strategies, one must exist in mixed strategies. A mixed strategy is a probability distribution over two or more pure strategies.
“One feature of mixed strategy equilibrium is that given the strategies chosen by the other players, each player is indifferent among all the actions that he or she selects with positive probability.” (Sethi, 2007, p.290). By this, we can see in the game matching pennies, given that player 2 chooses each action with probability one-half, player 1 is indifferent among choosing H, choosing T, and randomizing in any way between the two choices (Sethi, 2007, p.290).
Mixed strategy equilibrium in real-world
Mixed strategy nash equilibrium can be interpreted in different real-world situations. Here are some examples of mixed strategy in real-world:
- Playing a sport or a game like tennis is an example of mixed strategy. The player/s may be randomly deciding whether to serve the ball up the center or out the side of the court (Easley and Kleiberg, 2010, p. 178).
- A card player may be randomly deciding whether to bluff or not in a card game (Easley and Kleiberg, 2010, p. 178).
- Rock, paper, scissors is a game where two players have three strategies. This game is more difficult, but it has the same principle as mixed strategy equilibrium: make opponent indifferent between strategy choices (Faculty.econ.ucsb, year: unknown).
- Market entry game is also an example where mixes strategy can be use. For example:
Two firms must decide whether to put one of their restaurant in a shopping mall. The strategies are to “enter” or “don’t enter”. If both firms don’t enter, they get 0 profits. If one enters and the other don’t enter, the firm who entered earns 300000 dollars per year in profits (don’t enter gets 0 profit). If both decided to enter, both lose 100000 dollars per year as there is not enough demand to make positive profits. By this, we can check if it has mixed strategy equilibrium by calculating the probabilities (Faculty.econ.ucsb, year: unknown).
Easley, D and Kleinberg, J.(2010) Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press.
Garrett, K and Moore, E. (2008). International Review of Economics Education: Teaching mixed strategy nash equilibrium to undergraduates. Internet: https://www.economicsnetwork.ac.uk/iree/v7n2/garrett.pdf [read: 17.11.17]
Sethi, R. (2007). International Encyclopedia of the Social Sciences, 2nd edition: Mixed Strategy. Internet: http://www.columbia.edu/~rs328/MixedStrategy.pdf [read: 19.11.17]
Mixed Strategies. Internet: http://faculty.econ.ucsb.edu/~garratt/Econ171/Lect07and08_Slides.pdf [read: 19.11.17]