# Finitely repeated games, the end-of period problem, infinitely repeated games

Finitely Repeated Games

The first type of game that will be discussed in this Wiki is that of finitely repeated games. In situations such as this the game will only be repeated a finite number of times and will eventually end. There are two primary groups of repeated games; those in which the players know when they will end and those in which the players do not know when the game will end.

Uncertain Final Period
First, we will look at a game with an uncertain final period. This game will take into consideration two firms who can choose to either choose to sell their products at low prices or at high prices. The table below gives the game outcomes.
Table 1: A Pricing Game
 Firm B Price Low High Firm A Low 0, 0 50, -40 High -40, 50 10, 10
In a situation such as this, the choices made by the two firms will mirror those of firms in infinitely repeated games. In order for the companies to make a profit, they both will choose to go with high prices. While it does not give each company the highest return, it is a stable set of decisions. If one of the two firms were to change at any one period to the use of low prices, the firm will make a large amount of profit in that time period but the competing firm will begin to use low prices in the following period in order to keep from loosing such a large amount per game. Following this both companies will make zero profit. When the end period of the game is unknown the firm will not want to put itself at the risk of losing all profit, so it will continue to charge high prices indefinitely.

Known End Period
The second type of finitely repeated game is the game with a known end period. Since the end period is known in these games it gives rise to what is known as the end of period problem. To give an example of this we will look at the same game structure as is shown in table 1, but with only two periods for the game. With the lack of a third period, the two firms do not have the ability to punish the other for actions taken in the second period. Because of this, each firm will have an incentive to treat the game as a one shot version and as such both firms will go with the low prices in the second and final period of the game. Since each firm knows what the other will do in the final period, the period previous to this essentially becomes the final period. Due to this, each firm will again go with the low prices because they will be unable to punish the other in the following period. In fact, this type of behavior would continue even if there were far more periods simply because this type of logic by the two firms would be applied throughout the game and both firms would go with low prices for every period.

The End of Period Problem
The end of period problem can also be applied to other managerial style situations. As has been stated before, the problem with the final period is that if someone acts inappropriately they can not be punished in the following phase of the game. An example of this would be when a person announces that they are quitting their job. If the person will not be working the following day, then they have little incentive to put forth a large amount of effort today. A solution to this would be for the manager to fire the worker as soon as they announced their intention to quit. However, if this began happening then the workers would simply quit at the end of the day, giving no real notice to their manager. Since this strategy would not solve the end of period problem, a better solution as a manager would be to let the workers know that you are well connected and willing to write a letter of recommendation for them if need be. This still gives some power to punish those if they attempt to take advantage of the end of period problem, as a well connected manager could let others know about this workers previous exploits.

Infinitely Repeated Games

The other type of game that will be discussed here is the infinitely repeated game. In this situation the game is played over and over again forever. Once again, the game outcomes from table 1 will be used to give the example. As we have seen before, if this is a finitely played game with a known end date the two firms will both choose to charge low prices and as a result both will earn zero profits. However, if the game is to be played forever then some kind of collusion can often be reached between the two firms. Both firms will realize that they can maintain profits of 10 forever if they do not attempt to lower prices. However, if one firm lowers prices it will lead to both firms setting low prices and thus both firms making zero profit. Once this happens it will remain that way, as neither firm would want to incur a loss of 40 for the period. The other item to remember with this type of game is that the interest rate does make a difference in the firm deciding if it wishes to take the chance with the low prices. The firm must look at the present value of all cash flows from this game. If setting a low price and making a profit of 50 in the current period with zero profit for the rest of the game has a higher net present value than simply setting a high price for the duration of the game the firm may find it in their best interest to break away from the high prices.

Collusion
In order for the collusion between firms to work in the infinitely repeated games several factors need to be examined. The number of firms in the industry is an important item to watch, because as the number of firms increases it becomes more difficult for the various firms to collude. The differences in firm costs and demand is also important, as firms with wide variances in these areas will be unable to collude. Many companies may also feel inclined to cheat on collusion agreements due to the fact that large short term gains can often be found, even though it often leads to some form of punishment by the other firms in the industry. New entrants into the market may also cause issues with collusion, as a new firm may completely change the pricing structure of the industry.

Questions

1. The end of period problem can cause firms to treat a finitely repeated game with a known end date as what?
a. An infinitely repeated game
b. A one shot version of the game
c. A Nash equilibrium
d. A finite game with an uncertain final period

2. True or False. In an infinitely repeating game it may be profitable for a firm to take a large gain at the beginning of a game with zero profits in very period after that if the net present value of this option is greater than the net present value of the other options.

3. Which of the following are factors which affect collusion?
a. The number of firms in the industry
b. The differences in the sizes of the company
c. The temptation to cheat to gain advantages over other firms
d. All of the above

4. True or False. The end of period problem can be used to look at managerial issues such as workers quitting from their jobs.

5. A firm in a finitely repeated game with no known end date will act similarly to what other type of repeated game?
a. A finitely repeated game with known end date
b. A one shot game
c. An infinitely repeated game
d. None of the above