game can be defined with the following payoff matrix: In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%). Network traffic An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are "cars" traveling from to , what is the expected distribution of traffic in the network? This situation can be modeled as a "game", where every traveler has a choice of 3 strategies and where each strategy is a route from to (one of , , or ). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via experiences travel time of , where is the number of cars traveling on edge . Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, 100 cars are travelling from to , then equilibrium will occur when 25 drivers travel via , 50 via , and 25 via . Every driver now has a total travel time of 3.75 (to see this, note that a total of 75 cars take the edge, and likewise, 75 cars take the edge). Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via and the other 50 through , then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between and is removed,