An introduction about game theory

In this lecture we are going to explore how people make their choices when their outcome depends not only on their decisions but also on the ones of others. A situation when we have two or more people interacting and influencing the outcomes among each others is called a game. In order to explain the fundamental principles of game theory we will introduce the simplest game possible: a one shot simultaneous move game. In this kind of games two or more players can pick freely pick a single choice or strategy at the same time. A game like that is represented by a table where we represent on the outside separately the players and near them the different strategies available to them. Player 1 is represented to the left of the table while player 2 is above it. On the inside, are represented various cells which contain the outcomes if Player 1 and Player 2 pick the corresponding choices. Inside a cell, the first outcome refers to Player 1 and the second to player2.

HOW A GAME WORKS

Let’s suppose we have two companies: company A and company B who can both choose to launch a new product on the market. As said, in the first cell are represented the corresponding outcomes if both the companies launch the product, in the upper right cell there are the outcomes if company A launches the product while company B does not, in the third we refer to the case in which company B launches the product while company A does not and the last cell represents the outcomes if none of the companies launches the product. Now, in order to understand the choices of the companies we have to analyse them singularly. To determine which is the best choice for player 1 when player 2 behaves in a certain way, we have to consider each possible choice for player 2 individually and then analyse which is the best strategy for player 1 in any case meaning that we have to compare for each column (each column corresponds to a different choice for player 2) the various outcomes for player 1 and pick the highest one. Every time we determine the best strategy for a player we need to draw a circle or a line near the outcome. If company B decides to launch the new product, the best choice for it is to launch that product too while, if company B decides not to launch the product, the best choice for A is to launch the product. The same exact reasoning applies for player 2: we have to consider each possible choice of player 1 and determine which is the best choice for him in any possible scenario. Now we have to compare the possible outcomes in rows and no more in columns. If company A decides to launch the product, the best choice for company B is to do the same thing while, if company A decides not to launch the product, company B will be better off by launching it. Now notice that inside the matrix there is a cell where there are both the outcomes circled: the cell corresponding to launch-launch. That combination of strategies constitutes the Nash Equilibrium of the game. In a Nash equilibrium no player has an incentive to deviate from that strategy because otherwise he will be worse off.

A PECULIAR GAME: THE PRISONER’S DILEMMA

Not always the Nash Equilibrium represents the situation where both the collective interest is maximized. Look at the previous example, as we said, the players will in the end choose to follow the “launch” strategy but notice that if none of them launched the product, so in the cell corresponding to lot launch-not launch, both the players could be better off. A game where the players choose in the end a strategy that does not maximize collective interest is called a prisoner’s dilemma game. This term is given to it because of the classical example used to describe it: consider two prisoners who can either decide individually to cooperate or not with each other. If both cooperate they will both stay in jail for 2 years, if one of them cooperates while the other does not, the one who doesn’t will be free and the other will be arrested for 5 years, if none of them cooperates they will both be arrested for 3 years. By conducting the same analysis as before, we can find the Nash equilibrium of the game corresponding to the cell do not cooperate-do not cooperate. However it is clear that both of them could be better off by choosing to cooperate.

DOMINANT AND DOMINATED STRATEGIES

Notice that, in both the examples we made, players chose one strategy regardless of the choice of the other player meaning that their choice is not influenced by the counterpart. The strategy that is always chosen is called a dominant strategy while the strategy which is never chosen is called a dominated strategy. Identifying dominated strategies is useful when considering games where players can choose among more than two strategies. Consider once again company A and company B but this time let’s suppose that they can choose between not launching any product, launching one product or launching two products. In the matrix are represented the outcomes corresponding to their choices. As you can see, the strategy “launch two products” is never chosen by both the players. Because we are sure of this, that same strategy will not influence the final Nash equilibrium and therefore we can eliminate it and simplify the problem. By doing so, the 3x3 matrix is reduced to the 2x2 matrix we had in the first example. Obviously, when two players have a dominant strategy, the Nash equilibrium will be the combination of these strategies. If only one player has a dominant strategy, the Nash equilibrium can be found by choosing the best response for the player without it to the dominant strategy of the other player.

GAMES WITH MORE THAN ONE NASH EQUILIBRIUM

Consider two friends: Tom and Jerry who want to play videogames together; in order to do so, they have to buy the same console and they can either purchase a PlayStation or an Xbox. In the matrix is evident that the two guys will get a payoff only in the cells corresponding to PlayStation-PlayStation and Xbox-Xbox. Notice that there are two Nash equilibria. In that case the game does not present a solution meaning that there is no solution in pure strategies (pure strategies are the ones analysed until now where the result of the game is unique and clear). The game can still be solved but by using mixed strategies: strategies that depend on probabilities. Every game, even if it does not have a pure strategies solution surely has a mixed strategy solution. In our case we can suppose that both the guys will buy either console with probability ½. If this happens no one of the guys can do better by buying the console randomly with probability ½ for each choice and hope that the other has made the same choice. This game seems so trivial because we supposed equal probabilities and equal payoffs for both the players. In the example section you will find more difficult and meaningful exercises.

WHEN PLAYERS DO NOT CHOOSE AT THE SAME TIME

Until now we have considered games in which players chose their strategy at the same time and therefore are not really reacting to the other player’s strategy. If we let one player choose his strategy first, the game becomes a sequential move game. This kinds of games are represented by a game tree, a graph showing all the possible choices for the players ordered from the leftmost point which represents the choice of the player who plays first to the rightmost point which represent the final outcome if the players follow that combination of choices. For each of these combination of choices are represented the respective outcomes for both the players. This kind of games are solved thanks to a process called backward induction. To find the choices players will pick, we need to start from the rightmost point and determine which is the best choice for the player who is playing and then going backward on the tree until we reach the start point. Let’s consider as example the game we considered with company A and company B which can decide among launching 1 or 2 products or no product at all but, this time, it is company A who chooses first. Let’s start from the rightmost choice which is, in our example, the choice of company B. We have to identify its best choices for each possible choice of company A and mark them. We know for sure that company B will never choose one of the options which have not been marked and therefore we can eliminate them. Now, company A can choose one of the three strategy knowing what will the other player do to respond to each of them. Therefore company A will choose among the three the one which gives it the highest outcome. The Nash equilibrium this time is launch two products-do not launch. As you can see the equilibrium is much different from the original one.