Game Theory and Strategic Play in Economics
These are my notes and thoughts on game theory and strategic play in economics.
The simple economic framework we have developed thus far is not equipped to handl;e situations like these where your payoffs depend on the behavior of others and your behavior affects their payoffs. These situations include, among others, how to allocate scarce resources in partnerships, firms, friendships, and families. You may wonder what economics has to do with friendships and families. It has a lot to do with them.
There are important situations when the behavior of others affects your payoffs. Game theory is the economic framework that describes our optimal actions in such settings. A Nash equilibrium is a situation where none of the players can do better by choosing a different action or strategy for themselves. Nash equilibria are applicable to a wide variety of problems, including zero-sum games, the tragedy of the commons, and the prisoner’s dilemma.
Game theory is the study of situations in which the payoff of one agent depends not only on his or her actions, but also on the actions of others. It emerged as a branch of mathematics that first focused on the analysis of parlor games.
A payoff matrix represents the payoffs for each action players can take in a game. A best response is simply one player’s optimal strategy, taking the other player’s strategy as given.
When a player has the same best response to every possible strategy of the other player, then we say that the player has a dominant strategy. In the prisoner’s dilemma, confessing is a dominant strategy, because it is your best response to any strategy choice of your partner.
Interestingly, this equilibrium leads to an outcome that is not best for both players. Life doesn’t always present a game that has a dominant strategy.
This is the essence of the equilibrium concept proposed by John Nash, in equilibrium, no player in a game can change strategy and improve his or her payoff. Therefore, a combination of strategies is a Nash equilibrium if each player chooses a strategy that is a best response to the strategies of others, that is, players are choosing strategies that are mutual best responses.
This notion of equilibrium depends on two critical factors: that all players understand the game and the payoffs associated with each strategy and that all players understand that other players understand the game.
The key to finding Nash equilibria is to determine whether either player has an incentive to change his strategy. It is also a common occurrence in game theory to have more than one Nash equilibrium.
Game theory is most often used when a few players make choices that affect each other’s payoffs. The same type of reasoning applies even when the number of players is large.
A zero-sum game means that because one player’s loss is another’s gain, the sum of the payoffs is zero. Applying our method of finding Nash equilibria, we draw the arrows. They show that no Nash equilibrium exists, because no cell in the matrix has two arrows pointing in.
We say that a strategy is dominated if it yields lower payoffs than some other available strategy. It is a basic tenet of game theory that no player should pick a dominated strategy.
The games that we have discussed so far all revolve around two players choosing an action simultaneously. Suppose that, instead, one player goes first and the other chooses an action only after seeing how the first player chose. This type of situation, which specifies the order of play, is represented by an extensive-form game.
So we can say that an extensive-form game specifies the order of play and payoffs that will result from different strategies and uses a game tree to represent them.
The easiest way of approaching any extensive-form game is to use backward induction. Backward induction is the procedure of solving an extensive-form game by first considering the last mover’s decision. So, backward induction involves starting at the end of a game and solving it backwards.
A commitment is an action that one cannot take back. Commitments, which come before other actions, can change who has the advantage.
Game theory provides us with the tools to examine situations when players’ payoffs are intertwined. Whether decisions are made simultaneously or sequentially, game theory is about being able to see the world through the eyes of your opponent and understand the opponents incentives.
The key concepts of game theory are best responses and Nash equilibria. A best response is one agent’s optimal strategy taking the other player’s strategy as given. When the same strategy is the best response against any possible strategies of the other players, then it is a dominant strategy. In most games, players do not possess such a dominant strategy, making their best responses depend on the strategy choices of other players.
A Nash equilibrium arises if each player chooses a strategy that is a best response to the strategies of other players. Put differently, a Nash equilibrium is a combination of strategies that are mutual best responses.
The concept of Nash equilibrium enables us to make predictions about behavior in a range of situations, including those that can be modeled as the prisoner’s dilemma, the tragedy of the commons, and zero-sum games. It also helps us understand why trustworthy behavior is more likely to emerge when players have reputational concerns.
Suppose there are cable tv companies in your city, Astounding cable and Broadcast cable. They both must decide on a high advertising budget, a moderate advertising budget, or a low advertising budget. They will make their decisions simultaneously.
- Astounding dominant strategy is medium and Broadcast’s strategy is no dominant strategy
The equilibrium is:
- medium/medium
Consider the following game:
- David’s dominant strategy is low price and Jordan’s dominant strategy is low price
The Nash equilibrium is:
A strategy combination is a Nash equilibrium if each strategy is a best response to the strategies of others.
- D card
Chevron and BP are bidding against each other for new oil drilling leases in the Gulf of Mexico. The bids will be simultaneous with the higher bidder as the winner. Chevron decides to hire you as a consultant to help it use game theory to make the best decision on how much to bid. What elements must be known to set up a simultaneous move game?
- Players, strategies, and payoffs
What must you, as the consultant, construct for Chevron before you can determine if there is a dominant strategy equilibrium?
- Payoff matrix
After you examine the payoffs, you discern that BP’s best response is to always bid low. Bidding low would be BP’s:
- Dominant strategy
Suppose Russia is deciding to invade or not invade its neighbor Ukraine. The US has to decide to be tough or make concessions. They will make their decisions simultaneously. Their payoffs are as follows:
- US’s best response when Russia chooses to not invade is Tough
- US’s best response when Russia chooses to invade is Tough
- Russia’s best response when the US chooses to be tough is Not Invade
- Russia’s best response when the US chooses to make concessions is Invade
The Nash equilibrium is:
- Tough/Not Invade
It is possible for two player games to be quite asymmetric: Each player might have a different set of options, and the payoffs may be quite different. Consider the following example between a large firm and a small firm. The first number in each box denotes the large firm’s payoff, and the second number shows the small firm’s payoff.
In the given situation, neither of the firms has a dominant strategy. There is no dominant strategy because the best response of each player changes with the change in strategy chosen by the other player.
- Neither of the firms has a dominant strategy
In the given example, the Nash equilibria are:
- Medium price, expand operation and low price, stay small
Two gas stations, A and B, are locked in a price war. Each player has the option of raising its price or continuing to charge the low price. They will choose strategies simultaneously. If they both choose C, they will both suffer a loss of -25. If one chooses R and the other chooses C, the one that chooses R loses many of its customers and earns 0, and the one that chooses C wins many new customers and earns 250. If they both choose R the price war ends and they each earn 125. Does either player have a dominant strategy?
- The stations do not have dominant strategies because what works best depends on what the other station does
What is the Nash equilibrium?
- The Nash equilibria are for Station A to pick C and Station B to pick R and for Station A to pick R and Station B to pick C
Imagine a game in which two drivers drive toward each other on a collision course: one must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a chicken, meaning he is a coward.
If they both go straight, then they both die, earning -10 happiness points each. If one goes straight and the other swerves, then the brave driver gets 5 points of happiness and the chicken loses 2 points of happiness. If both drivers swerve, then it is a tie and nobody earns any happiness points.
- -10/-10 5/-2
- -⅖ 0/0
Driver 1 does not have a dominant strategy and Driver 2 does not have a dominant strategy
Identify the Nash equilibrium?
- Boxes 2 and 3
Suppose that a player has a dominant strategy. Would she choose to play a mixed strategy?
- No, because it would involve choosing actions other than the dominant strategy
Use a matrix to model a two-player game of rock-scissors-paper with a payoff of 1 if you win, -1 if you lose, and 0 if you tie
- p1/p2 Rock Paper Scissors
- Rock 0/0 -1/1 1/-1
- Paper 1/-1 0/0 -1/1
- Scissors -1/1 1/-1 0/0
The Nash equilibrium is:
- There is no Nash equilibrium
Why should you use a mixed strategy to play this game?
- Predictable behavior by one player can be taken advantage of by the other player
A pure strategy involves
- Choosing one particular action for a situation
Suppose that a goalie is playing a mixed strategy between diving to the left and the right. A player decides which strategy to employ when playing a game with mixed strategies by choosing:
- Randomly
In a game with mixed strategies, does either of the players have a dominant strategy?
- No, because the best choice in a mixed strategy game is to pick a random strategy
The game is between two players, player A and player B. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, player A keeps both pennies. If the pennies do not match player B keeps both pennies.
1/-1 -1/1
-1/1 1/-1
- Player A does not have a dominant strategy and player B does not have a dominant strategy
This payoff matrix is an example of a
- zero-sum game
Identify the Nash equilibriums:
- There is no Nash equilibrium
A first-mover advantage occurs if
- The first mover to act in a sequential game gets a benefit from doing so
Suppose you were playing rock-paper-scissors as an extensive form game; first you choose rock, or paper, or scissors, and then your opponent makes a choice. Is there a first-mover advantage in this game?
- No, if you show your move first you will lose every time
Two firms are thinking of entering a new market. If one enters it will be successful but if a second enters both will suffer very large losses. Is there a first-mover advantage in this game?
- Yes, the firm that goes first can enter and the firm that goes second will have no incentive to enter
Consider a game with two players, Bill and Larry. They play the game as summarized in the game tree below. Suppose Bill has a secure monopoly in the market for computer operating systems. When there is no threat of entry, Bill charges the maximum monopoly price. Then Bill discovers that a second software firm run by Larry is thinking about entering the market. Since his monopoly is now insecure, Bill has two options; he can continue to act as a monopolist and allow the second software firm to enter the market, or he can try to prevent the other firm from entering the market by producing a greater quantity at a lower cost.
If Bill acts as a secure monopoly and produces a low quantity at a high price, then Larry:
- Will enter
If Bill acts as an insecure monopoly and produces a greater quantity at a lower price, then Larry:
- Stay out
The logical move for Bill is to produce:
- High quantity
Assuming Bill acts logically, then the logical move for Larry is to:
- Stay out
Assuming Bill and Larry act logically, then Bill’s profit will be:
- 1000 and Larry’s profit will be 0
This is an example of:
- Extensive game
One of your coworkers, Freddie, really gets on your nerves and it would dramatically increase your happiness if he was fired. Freddie also dislikes you and wishes you would be fired. You have the option of going to your boss and lying about Freddie harassing you, which you are sure will get him fired. Freddie has the same option to make up a lie about you to get you fired. Whoever gets fired will not have any credibility to have his lies believed afterward.
Given these circumstances, it would be logical for you to:
- lie first to get Freddie fired
The person who lies first in this situation has a:
- First-mover advantage
Now assume that your coworkers will know if either you or Freddie lie to get the other fired, and your coworkers will shun the liar and make life miserable for that person.
Given these circumstances:
- Do nothing
The person who lies first in this situation:
- Suffers reputational damage
Suppose there is a very inspirational public figure who decides to run for president. This person makes a number of great sounding promises to the people about how he will improve the country if he becomes president. The people are excited about the politician’s promises, but realize there is a chance that the politician is a liar and will sell out to specific interest groups after he is elected and not keep his promises.
Based on this:
- Extensive form game, the people should not elect this politician
Suppose that a new idealistic political party publicly promises before the election that if the politician sells out after he is elected, then it would use its considerable power and influence to impeach the politician and throw him in prison.
If this threat is credible, then the people:
- Should elect the politician because the commitment of the new political party changes the payoff the politician faces
The police confront Snoof and Charlie with the payoff matrix below.
Snoop’s dominant strategy is to confess and Charlie’s dominant strategy is to confess.
The Nash equilibrium is:
- Box 4