Poker and gambling are popular examples of zero-sum games since the sum of the amounts won by some players equals the combined losses of the others. Games like chess and tennis, where there is one winner and one loser, are also zero-sum games. The game of matching pennies is often cited as an example of a zero-sum game, according to game theory. The game involves two players, A and B, simultaneously placing a penny on the table.
The payoff depends on whether the pennies match or not. As can be seen, the combined playoff for A and B in all four cells is zero. Zero-sum games are the opposite of win-win situations—such as a trade agreement that significantly increases trade between two nations—or lose-lose situations, like war, for instance. In real life, however, things are not always so obvious, and gains and losses are often difficult to quantify. In the stock market, trading is often thought of as a zero-sum game.
However, because trades are made on the basis of future expectations, and traders have different preferences for risk, a trade can be mutually beneficial. Investing longer term is a positive-sum situation because capital flows facilitation production, and jobs that then provide production, and jobs that then provide savings, and income that then provides investment to continue the cycle. Game theory is a complex theoretical study in economics.
Game theory is the study of the decision-making process between two or more intelligent and rational parties. Game theory can be used in a wide array of economic fields, including experimental economics , which uses experiments in a controlled setting to test economic theories with more real-world insight.
When applied to economics, game theory uses mathematical formulas and equations to predict outcomes in a transaction, taking into account many different factors, including gains, losses, optimality, and individual behaviors. When applied specifically to economics, there are multiple factors to consider when understanding a zero-sum game. Zero-sum game assumes a version of perfect competition and perfect information; both opponents in the model have all the relevant information to make an informed decision.
Taking a step back, most transactions or trades are inherently non-zero-sum games because when two parties agree to trade they do so with the understanding that the goods or services they are receiving are more valuable than the goods or services they are trading for it, after transaction costs.
These games involve only two players; they are called zero-sum games because one player wins whatever the other player loses. Consider the simple game called odds and evens. Suppose that player 1 takes evens and player 2 takes odds.
Then, each player simultaneously shows either one finger or two fingers. Hopefully, you are beginning to see some of the challenges for analyzing non-zero-sum games. We know there are equilibrium points in Battle of the Sexes, but even rational play may not result in an equilibrium. For the remainder of this section, let's assume that players are not allowed to communicate about strategy prior to play.
Such games are called non-cooperative games. Before moving on, let's try to find the maximin strategies for our players using the graphical method, as we did with zero-sum games.
Consider Battle of the Sexes from Bob's point of view. We know that Bob wants to maximize his payoff that has not changed. So Bob doesn't care what Alice's payoff's are. Write down Bob's payoff matrix. Recall that the graphical method represents Bob's expected payoff depending on how often he plays each of his options. Sketch the graph associated with Bob's payoff matrix.
The area between the two lines still represents the possible expected values for Bob, depending on how often Alice plays each of her strategies. So as before, the bottom lines represent the least Bob can expect as he varies his strategy.
Thus, the point of intersection will represent the biggest of these smallest values the maximin strategy. Find this point of intersection. How often should Bob play each option? What is his expected payoff?
So no matter what Alice does, Bob can expect that over the long run he wins at least the value you found in Exercise 4. Make sure you understand this before moving on. Now consider Battle of the Sexes from Alice's point of view.
Write down her payoff matrix and use the graphical method to find the probability with which she should play each option and her expected payoff.
Now, from Exercise 4. Note that since this is not a zero-sum game, both players can expect a positive payoff. But now we want to see how this pair of mixed strategies really works for the players. Assume Bob plays the mixed strategy from Exercise 4. Are Alice's expected values equal? That is, there is no single optimal strategy that is preferable to all others, nor is there a predictable outcome. Non-zero-sum games are also non-strictly competitive, as opposed to the completely competitive zero-sum games, because such games generally have both competitive and cooperative elements.
Players engaged in a non-zero sum conflict have some complementary interests and some interests that are completely opposed. The Battle of the Sexes is a simple example of a typical non-zero-sum game. In this example a man and his wife want to go out for the evening. They have decided to go either to a ballet or to a boxing match. Both prefer to go together rather than going alone.
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