Game Theory example

Why must each firm in an oligopoly act strategically?

Because its profit not only depends on how much output it produces, but also on how much *other firms produce as well*.

What is the Pittsburgh Left-Turn Game?

A game of two equilibria wherein the government steps up to impose one equilibrium by law in order to avoid accidents .

What is understood by mixed strategy?

A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move. A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player.

What is meant by sequential games?

In sequential games, a series of decisions are made in a particular sequence of which the outcome affects successive possibilities. 1: The analysis of sequential games is of great interest because they usually model reality better than simultaneous games: producers will usually observe demand before deciding how much output to produce, duopolists will observe each other's decisions before dumping more goods in the market, etc.

What is understood by repeated games?

A form of game theory in which some base game is repeated. It captures the idea that a player will have to take into account the impact of his current action on the future actions of other players; this is sometimes called his reputation. Two strategies apply: 1: *Follow the agreement if the other player follows it*. 2: *Deviate from the agreement if the other player deviates*. Repeated games can sustain cooperation if the short run profits from deviating are smaller than future losses due to relation. --- Repeated games introduce a new series of incentives: the possibility of cooperating means that we may decide to compromise in order to carry on receiving a payoff over time, knowing that if we do not uphold our end of the deal, our opponent may decide not to either. Our offer of cooperation or our threat to cease cooperation has to be credible in order for our opponent to uphold their end of the bargain. Working out whether credibility is merited simply involves working out what weighs more: the payoff we stand to gain if we break our pact at any given moment and gain an exceptional, one off payoff, or continued cooperation with lower payoffs which may or may not add up to more over a given time. Therefore, each player must consider their opponent's possible punishment strategies.

What is the prisoner's dilemma?

A particular game between two captured prisoners that illustrates why cooperation is difficult to main even when it is mutually beneficial.

What is understood by a pure strategy?

A pure strategy defines a specific move or action that a player will follow in every possible attainable situation in a game. Such moves may not be random, or drawn from a distribution, as in the case of mixed strategies. A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face. A player's strategy set is the set of pure strategies available to that player.

What is meant by simultaneous games?

A simultaneous game is one in which all players make decisions (or select a strategy) without knowledge of the strategies that are being chosen by other players. 1: Even though the decisions may be made at different points in time, the game is simultaneous because each player has no information about the decisions of others; thus, it is as if the decisions are made simultaneously. 2: Simultaneous games are represented by the normal form and solved using the concept of a Nash equilibrium.

What is understood by the Nash equilibrium?

A situation in which agents interacting with one another each choose their best strategy given the strategies that all the other actors have chosen, *such that no player can improve his payoff by independently changing his strategy*.

What is the volunteer's dilemma?

A situation in which each of a number of players faces the decision of either making a small sacrifice from which all will benefit, or free-riding. 1: If no one volunteers, the worst possible outcome is obtained for all participants. If any one person elects to volunteer, the rest benefit by not doing so. 2: Because the volunteer receives no benefit, there is a greater incentive for free-riding than sacrificing oneself for the group. If no one volunteers, everyone loses.

What is understood by a dominant strategy?

A strategy that is best for a player in a game regardless of the strategies chosen by the other players.

What is understood by a sub-game perfect Nash equilibrium?

An equilibrium such that players' strategies constitute a Nash equilibrium in every subgame of the original game. It may be found in sequential games by backward induction. 1: First, one determines the optimal strategy of the player who makes the last move of the game. 2: Then, the optimal action of the next-to-last moving player is determined taking the last player's action as given. 3: The process continues in this way backwards in time until all players' actions have been determined.

Why is cooperation and reaching the optimal equilibrium difficult?

Cooperation between the Bonnie and Clyde is difficult to maintain, because cooperation is individually irrational.

What are the type of games?

Depends on a number of factors: 1: Whether agents take actions at the same time (*simultaneous game*) or in turns (*sequential game*). 2: Whether agents have access to the same information. 3: In which way the action of one agent affects the payoff of the other agent. 4: Whether the game is played only once or several times.

What is understood by rationality?

It is assumed that each player seeks to *maximise their payoffs*. It is assumed that each player is able to *calculate the actions* which will maximise their payoffs. It is assumed that every player *understands that every player is rational*.

Why do Iran and Saudi Arabia not reach the optimal outcome of 50 billion each?

Self interest makes it difficult for the oligopolists to maintain the cooperative outcome with low production, high prices and monopoly profits.

What is understood by "strategic" situations?

Situations in which each person, in deciding what actions to take, must consider how others might respond to that action.

What are the assumptions of game theory?

1: *Rationality*. 2: *Full knowledge of the game*.

What are the four elements of game theory?

*Players*: a game must specify the players of the game. *Strategy*: a game must specify the strategies available to each player at each decision point. *Information*: a game must specify the information available to each player at each decision point. *Payoffs*: a game must specify the payoffs for each outcome.

Give an example of a cooperative cartel outcome. What are the implications?

1: *Dominant strategy* for both Jack and Jill is the have high production (40), because this is the best strategy regardless of whether the other player chooses to produce high or low. 2: *Optimal equilibrium*, however, is reached when both Jack and Jill have low production (30). If Jack and Jill cooperate they will reach an outcome equal to a monopoly outcome (60). However, as we've seen previously, both Jack and Jill will be tempted to produce more and reap higher profits. The outcome if the game is played once is thus that both players run with their dominant strategy and gain lower profits.

Give an example of a prisoner's dilemma in oil production between two oligopolists.

1: *Dominant strategy* for both players is high production. 2: *Optimal equilibrium* is for both to produce low and get 50 billion each. 3: *Nash equilibrium* is reached when both countries follow their dominant strategy - produce high and get 40 billion each. Outcome: (high production, high production) Payoff: (40 billion, 40 billion)

Give an example of a prisoner's dilemma in an arms race between the US and the USSR.

1: *Dominant strategy* for both the US and USSR is to arm and gain geopolitical influence. 2: *Optimal equilibrium* is reached when both players disarm and peace ensues. 3: *Nash equilibrium* is reached when both players follow their dominant strategy, because that is the best response to the strategies of the other player. Thus, a lack of cooperation in an arms race is bad for everyone.

Give an example of the prisoner's dilemma.

1: Bonnie and Clyde have been captured. Police have enough evidence to charge them on a weapons charge for 1 year, but suspect that they have been involved a bank robbery which gives 20 years. Because they lack hard evidence, they need one to confess. 2: The police questions Bonnie and Clyde in separate rooms. a. If one confesses he get immunity, while the other gets 20 years. b. If both confesses they both get 8 years. c. If both remain silent they both get 1 year.

What is the equilibrium outcome in case of dominant strategies? What is the equilibrium outcome with no dominant strategies?

1: If there is a dominant strategy in a simultaneous move game, the outcome will be a situation in which *each player plays his dominant strategy*. 2: If there is no dominant strategy we find a combination of strategies such that each player's strategy is a *best response given the strategies of the other players*, then we have found an equilibrium of the game.

What is understood by a game with no pure Nash equilibrium?

A game where no pure Nash equilibrium exists. Consider the matching pennies game: 1: There is no (pure strategy) Nash equilibrium in this game. If we play this game, we should be "unpredictable." That is, we should randomize (or mix) between strategies so that we do not get exploited. 2: But not any randomness will do: Suppose Player 1 plays .75 Heads and .25 Tails (that is, Heads with 75% chance and Tails with 25% chance). Then Player 2 by choosing Tails (with 100% chance) can get an expected payoff of 0.75×1 + 0.25×(-1) = 0.5. But that cannot happen at equilibrium since Player 1 then wants to play Tails (with 100% chance) deviating from the original mixed strategy. 3: Since this game is completely symmetric it is easy to see that at *mixed strategy Nash equilibrium* both players will choose Heads with 50% chance and Tails with 50% chance. 4: In this case the expected payoff to both players is 0.5×1 + 0.5×(-1) = 0 and neither can do better by deviating to another strategy (regardless it is a mixed strategy or not).

What is an example of the volunteer's dilemma?

A group of N people including you are standing on a riverbank and observe that a stranger is drowning in the cold river. Do you jump in to save him or do you stay out? 1: Everyone benefits from the person surviving. 2: No one wants to sacrifice himself. 3: The primary objective is thus to free-ride, where the person survives without sacrificing yourself. Following a satisfaction scale (from -10 to 5), you rank each of the possible outcomes as follows: 5 = no sacrifice, person survives (free-ride) -1 = sole sacrifice, person survives 0 = sacrifice with others, person survives -10 = no sacrifice, person dies We see that there *is no dominant strategy*. 1: If you jump in the river and others jump in, you would have rather stayed out. 2: If you stay out and others stay out, you would have rather jumped in. 3: The primary objective is to free-ride.

What is the Nash equilibrium reached by Bonnie and Clyde?

In Nash equilibrium, each player is playing the strategy that is a best response to the strategies of the other players. No one has an incentive to change his strategy given the strategy of the others. Thus, Nash equilibrium is reached when both Bonnie and Clyde confesses.

What is the outcome of a finitely repeated prisoner's dilemma?

In a one-shot prisoner's dilemma, Nash equilibrium is reached at (confess, confess). In a finitely repeated game (when the prisoners know the number of repetitions) one solves the game by backward induction: 1: Consider the strategies of each player when they realise the next round is going to be the last - they behave as if it was a one-shot game, thus the (confess, confess) Nash equilibrium applies. 2: Now consider the game before the last. Since each player knows in the next, final round they are going to confess, there's no advantage to denying on this round either. The same logic applies for prior moves. Therefore, (confess, confess) is the Nash equilibrium for all rounds.

What is the outcome of an infinitely repeated prisoner's dilemma?

In a one-shot prisoner's dilemma, Nash equilibrium is reached at (confess, confess). In an infinitely repeated game, the outcome is different from both a one-shot game and a finitely repeated game - there will be no last round, and so a backwards induction reasoning does not work here. 1: At each round, both prisoners reckon there will be another round and therefore there are always benefits arising form the cooperation (denying). 2: However, prisoners must take into account punishment strategies, in case the other player confesses in any round.

How does the prisoner's dilemma relate to the welfare of society?

In some cases, the noncooperative equilibrium is bad from society's standpoint. 1: In the arms race example, both countries end up at risk. 2: In the common resources game, the extra wells dug are wasteful. 3: However, in the case of a cartel trying to maintain monopoly profits, the noncooperative solution is an improvement from the standpoint of society.

What is understood by full knowledge of the game?

It is assumed that each player *knows the rules of the game*. It is assumed thus that *payoffs and actions are observable and known by all*.

Give an example of a sequential game.

In this game firm 1 must choose whether to compete in a monopolistic market or not. 1: As there are no existing competitors in the market, should they choose not to enter, its payoff will be zero and the existing monopoly will have a payoff of two. 2: However, if firm 1 chooses to enter we reach node two, where the second decision must be made by the challenged player (the monopoly), whether to accommodate the new competitor or fight back to try and block its entry....If they choose to fight, both players will have a payoff of -1, whereas if the monopoly decides to accommodate the new player, our monopoly becomes a duopoly and the resulting drop in prices creates larger market demand, meaning that the existing company will still receive a payoff of one and the new player a payoff of two. 3: We assume that both players dispose of complete and perfect information. They have both carried out a thorough market study and know the consequences of each outcome. In that case, what they can do is work backwards to know what the competition will do. If firm 1 enters the market, player 2 has no reason to fight back, as its payoff will suffer by two points. This means that player 2 knows what the outcome will be if they choose to enter, so all they have to do is choose between a payoff of two if the new firm enters the market or zero if it doesn't

Lack of cooperation is a problem for those involved in the game, but how does it affect society as a whole?

It depends... 1: In a prisoner's dilemma, a lack of cooperation is bad for criminals but good for society (Nash equilibrium results in longer sentences). 2: In an oligopoly, a lack of cooperation is bad for the oligopolists but good for society (Nash equilibrium results in a higher quantity and lower price). 3: In an arms race, a lack of cooperation is bad for everyone (Nash equilibrium results in arming, which jeopardises peace). 4: In an effort to reduce global warming, a lack of cooperation is bad for everyone (Nash equilibrium is reached when no one reduces emissions).

What is an example of an oligopolistic prisoner's dilemma?

Jack and Jill are trying to keep the sale of water low to keep the price high. After reaching an agreement, each person must decide whether to follow the agreement. The dominant strategy for Jack is to *produce at a high rate*... 1: If Jill produces at a high rate, Jack will earn a higher amount of profit if he too produces at a high rate. 2: If Jill produces at a low rate, Jack will earn a higher profit if he produces at a high rate as well. The dominant strategy for Jill is to *produce at a high rate*..... 1: If Jack produces at a high rate, Jill will earn a higher amount of profit if he too produces at a high rate. 2: If Jack produces at a low rate, Jill will earn a higher profit if he produces at a high rate as well. Even though total profit would be highest if both individuals produced at a low rate, self-interest will encourage them to produce at a high rate.

Give an example of a repeated game.

Jack and Jill now has to take into account the impact of their current strategic decisions on the future actions of the other. 1: If Jack and Jill may agree to produce at the monopoly output, which is 30L each for a profit of 1800 each. 2: If Jack and Jill cooperate (in case of infinite periods) their payoff is 1800 forever. 3: However, (in case of infinite periods) if either Jack or Jill breaks the cartel agreement and produces 40L, they will both produce 40L and have payoffs of 1600 forever. 4: In case of a last period scenario, Jack or Jill now has an incentive to cheat because they will make a profit of 2000 without needing to worry about future periods where they would make 1600. 5: *It is therefore only rational for each farmer to cooperate if there is at least one future period*. Repeated games can thus sustain cooperation if the short run profits (2000) from deviating are smaller than future losses (infinite 1600) due to relation.

Tim and Greg are playing tennis. Every point comes down to whether Greg guesses correctly whether Tim will hit the ball to Greg's left or right. The outcomes are: Does either player have a dominant strategy? If Tim chooses a particular strategy (Left or Right) and sticks with it, what will Greg do? So, can you think of a better strategy for Tim to follow?

Neither player has a dominant strategy in this game. Jeff should hit left if Steve guesses right and Jeff should hit right if Steve guesses left. Steve should guess left if Jeff hits left and Steve should guess right if Jeff hits right. Thus, if Jeff stuck with a particular strategy (left or right), Steve would be able to guess it easily after a few points. A better strategy for Jeff is to randomly choose whether to hit the ball left or right, sometimes hitting left and other times hitting right.

Is Nash equilibrium necessarily the best possible outcome (optimal equilibrium)?

No, if Bonnie and Clyde had remained silent, they would have been better of collectively (optimal equilibrium). By each pursuing his or her own self-interests, the prisoners together reach an outcome that is worse for both of them, as they get 8 years instead of 1. Nash equilibrium is thus worse for the criminals, but better for society.

Give two examples other than oligopoly to show how the prisoners' dilemma helps to explain behaviour.

The arms race and common resources are some examples of how the prisoners' dilemma helps to explain behavior. In the arms race during the Cold War, the United States and the Soviet Union could not agree on arms reductions because each was fearful that after cooperating for a while, the other country would cheat. When two companies share a common resource, they would be better off sharing it. But, fearful that the other company will use more of the common resource, each company ends up overusing it.

What are the dominant strategies for Bonnie and Clyde can be made?

The dominant strategy is the strategy that is best for Bonnie or Clyde regardless of the strategy chosen by the other. *Bonnie's dominant strategy is to confess*: 1: If Clyde remains silent, Bonnie can go free for confessing. 2: If Clyde confesses, Bonnie can lower her sentence by confessing. *Clyde's dominant strategy is to confess*: 1: If Bonnie remains silent, Clyde can go free by confessing. 2: If Bonnie confesses, Clyde can lower his sentence by confessing.

Give an example of a prisoner's dilemma in advertising between Marlboro and Camel.

The options are either to advertise or not to advertise. a. If both don't advertise, they split the market in half. b. If both advertise, they split the market in half but profits are lower due to the cost of advertising. 1: *Dominant strategy* for both is to advertise because that is the best strategy regardless of the strategies chosen by the other player. 2: *Optimal equilibrium* is reached when neither player advertises because each earns higher profits (due to no advertising costs). 3: *Nash equilibrium* is reached when both players follow their dominant strategy. Thus, a lack of cooperation is bad for firms.

Give an example of a prisoner's dilemma in drilling between Chevron and Exxon.

The options are to drill one or two wells. If both players drill one well, the market is split in half. If both players drill two wells, the market is split in half but profits are lower due to the cost of drilling the wells. 1: *Dominant strategy* for both players is to drill two wells because that is the best strategy regardless of whether the other player drills one or two wells. 2: *Optimal equilibrium* is reached when both players drill one well because this maximises profits and splits the market equally. 3: *Nash equilibrium* is reached when both players follow their dominant strategy. Thus, lack of cooperation is bad for firms.

What is an example of a simultaneous game?

The prisoner's dilemma. 1: Each prisoner has *complete information* in that they know what they stand to win or lose. If both deny they get minimum term, if one denies and the other confesses the confessor goes free, etc. The rules of the game and each player's payoffs are common knowledge. 2: Each prisoner must also anticipate what the other prisoner does, as well as take into account that the other prisoner will do this in turn. 3: The Nash equilibrium is reached when both prisoners arrive at a rational decision that they have no reason to change - whatever else they do, they will only be worse off. In the prisoner's dilemma, this equilibrium is (confess, confess). 4: This situation can only be improved if the other prisoner chooses to do something else. However, we know that this won't occur in the prisoner's dilemma because confessing is the dominant strategy for each prisoner.

What is the prisoners' dilemma, and what does it have to do with oligopoly?

The prisoners' dilemma is a game between two people or firms that illustrates why it is difficult for opponents to cooperate even when cooperation would make them all better off. Each person or firm has a great incentive to cheat on any cooperative agreement to make himself or itself better off. Thus, firms have a difficult time maintaining a cooperative agreement.

What is game theory?

The study of how people behave in strategic situations.

Why do players sometimes cooperate even though it is rational for them to cheat?

While cooperation is difficult to maintain because of its irrationality, it is not impossible: Cooperation is easier to enforce *if the game is repeated*.

Farmer Jones and Farmer MacDonald graze their cattle on the same field. If there are 20 cows grazing in the field, each cow produces €4 000 of milk over its lifetime. If there are more cows in the field, then each cow can eat less grass, and its milk production falls. With 30 cows on the field, each produces €3 000 of milk; with 40 cows, each produces €2 000 of milk. Cows cost €1 000 apiece. a. Assume that Farmer Jones and Farmer MacDonald can each purchase either 10 or 20 cows, but that neither knows how many the other is buying when she makes her purchase. Calculate the pay-offs of each outcome. b. What is the likely outcome of this game? What would be the best outcome? Explain. c. There used to be more common fields than there are today. Why? (For more discussion of this topic, reread Chapter 11.)

a. If Jones has 10 cows and Smith has 10 for a total of 20 cows, each cow produces $4,000 of milk. Because a cow costs $1,000, profits would be $3,000 per cow, or $30,000 for each farmer. If one farmer had 10 cows and the other farmer had 20 cows for a total of 30 cows, each cow produces $3,000 of milk. Profits per cow would be $2,000, so the farmer with 10 cows makes $20,000; the farmer with 20 cows makes $40,000. If both farmers have 20 cows for a total of 40 cows, each cow produces $2,000 of milk. Profit per cow is $1,000, so each farmer's profit is $20,000. The results are shown in the table: b. If Jones had 10 cows, Smith would want 20 cows. If Jones had 20 cows, Smith would be indifferent (get the same profit) if he had 10 or 20 cows. So Smith has a dominant strategy of having 20 cows. If Smith had 10 cows, Jones would want 20 cows. If Smith had 20 cows, Jones would be indifferent (get the same profit) if he had 10 or 20 cows. So Jones has a dominant strategy of having 20 cows. The Nash equilibrium is for each farmer to have 20 cows, because that is the dominant strategy for each. They each make profits of $20,000. But they would both be better off if they cooperated and each had only 10 cows; then profit would be $30,000 each. c. The problem illustrates how a common field may be overused, reducing the profits of producers. Because people tend to overuse common fields, it is more efficient for people to own their own portion of the field. Thus, over time, common fields have been divided up and owned privately.

Little Kona is a small coffee company that is considering entering a market dominated by Big Brew. Each company's profit depends on whether Little Kona enters and whether Big Brew sets a high price or a low price: Big Brew threatens Little Kona by saying, 'If you enter, we're going to set a low price, so you had better stay out.' Do you think Little Kona should believe the threat? Why or why not? What do you think Little Kona should do?

a. If Kona enters, Big Brew would want to maintain a high price. If Kona does not enter, Big Brew would want to maintain a high price. Thus, Big Brew has a dominant strategy of maintaining a high price. If Big Brew maintains a high price, Kona would enter. If Big Brew maintains a low price, Kona would not enter. Kona does not have a dominant strategy. b. Because Big Brew has a dominant strategy of maintaining a high price, Kona should enter . c. There is only one Nash equilibrium. Big Brew will maintain a high price and Kona will enter. d. Little Kona should not believe this threat from Big Brew because it is not in Big Brew's interest to carry out the threat. If Little Kona enters, Big Brew can set a high price, in which case it makes $3 million, or Big Brew can set a low price, in which case it makes $1 million. Thus the threat is an empty one, which Little Kona should ignore; Little Kona should enter the market. e. If the two firms could successfully collude, they would agree that Big Brew would maintain a high price and Kona would remain out of the market. They could then split a profit of $7 million.

Assume that two airline companies decide to engage in collusive behaviour. Let's analyse the game between two such companies. Suppose that each company can charge either a high price for tickets or a low price. If one company charges €100, it earns low profits if the other company charges €100 also, and high profits if the other company charges €200. On the other hand, if the company charges €200, it earns very low profits if the other company charges €100, and medium profits if the other company charges €200 also. a. Draw the decision box for this game. b. What is the Nash equilibrium in this game? Explain. c. Is there an outcome that would be better than the Nash equilibrium for both airlines? How could it be achieved? Who would lose if it were achieved?

a. Decision box is as follows. b. If Braniff sets a low price, American will set a low price. If Braniff sets a high price, American will set a low price. So American has a dominant strategy to set a low price. If American sets a low price, Braniff will set a low price. If American sets a high price, Braniff will set a low price. So Braniff has a dominant strategy to set a low price. Because both have a dominant strategy to set a low price, the Nash equilibrium is for both to set a low price. c. A better outcome would be for both airlines to set a high price; they would both get higher profits. That outcome could only be achieved by cooperation (collusion). If that happened, consumers would lose because prices would be higher and quantity would be lower .