Game study is the study of strategic interaction where one player’s decision depends on what the other player does. What the opponent does also depends upon what he thinks the first player will do.

- Dominant strategy – when one choice gives better result than other
- Nash equilibrium – where each player has nothing to gain by changing strategy, given the choices of the other player. A Nash equilibrium is not necessarily pareto efficient. Both players could gain from co-operation.

## Examples of Game Theory

### Price war

This is a similar outcome but for two firms that can keep prices high and stable or start a price war. The best outcome for both firms is (a) $40, $40.

However, when prices are stable, if one firm cuts prices (starts price war) it will see profits rise to $60. However, the other firm who keeps prices high will lose market share and get zero profits. Therefore, the firm who loses out will almost certainly retaliate and the outcome will move to (d) with both firms just making $3 profit. Therefore, there is strong incentive to avoid price war.

- Co-ordination playoff
- In this example, if neither firms invest, they will make $50 each. However, if they both invest in new technology, which will become new market standard, they will both get substantially better pay off (a) with $200 each.
- However, if one firm invests in new technology and the other doesn’t, then they will be left with $0 (it is not widely shared). In this case, the firm will probably start investing too, as they would be better off.
- However, the key thing is whether one firm is willing to take the plunge and make zero profits in the short-run. It may not be able to afford this outcome.
- The issue with this game theory dillema is that there are strong rewards from co-operating. But, in the real world, for various reasons, co-operation may not be there.

#### Matching pennies

- This is a game with two players. They both put a penny on the table.
- If the pennies are Heads/heads or tails/tails – then Player A wins both pennies. He gains 1, (player B loses 1)
- If the pennies are mixed (heads/tails) or tails/heads then play B wins both pennies.
- This is an example of a zero-sum game – the net benefit is always zero. For everyone who gains, there is an equal and opposite loss.

#### Zero-sum game

In this situation, we have another zero-sum game situation. If a firm enters or leaves, there is always a net benefit of zero.

For firm A, its dominant strategy is to enter the market, because 1 is greater than -2.

For firm B, its dominant strategy is also to enter the market because -1 is greater than -3. Firm B would prefer both firms to leave the market so it can get to zero. But, in this model, it can’t do that because it know if A enters, it will have to enter or face the costs of -3.

#### Tariff or trade war?

- In this case, if both countries, pursue low tariffs, the outcome is £3m net welfare for each country. If A places tariff, then its net welfare will be £2m, and country B who keeps low tariffs will make £1.5m.
- If B retaliates and places tariffs on too, it will make itself worse of – welfare falls to £1m, but it will effectively punish A whose welfare falls from £2m to £1m.
- If firms wish to maximise welfare, they would stick to low tariffs. That is their dominant strategy and nash equilibrium.
- However, in the real world, there may be political pressures (e.g. protect domestic industry, even at expense of higher prices for consumers, which encourages countries to place tariffs.

#### Prisoner’s dilemma

#### The prisoner’s dilemma is a classic example of game theory.

- There are two prisoners held in solitary confinement. They can either confess to crime or stay silent (not confess)
- If both stay silent, they both get light sentence of 1 year.
- If they both confess, they get 5 years each.
- However, if one confesses to the crime and betrays the other, then the one who confesses is given immunity for giving information. But the other who remained silent gets 20 years.
- Therefore, a prisoner would only choose to remain silent, if they can guarantee the other prisoner will remain silent.
- The dominant strategy for both players is to confess. At worst they will get 5 years, at best they will get 0 years.
- The Nash equilibrium is confess/confess (5 years each). Because if a player acted unilaterally, it would be worse off.

### Decision Tree

Another way of describing game theory is through a decision tree.

- In this example, Firm A can choose to enter or leave. Firm B (the incumbent can then decide to fight (cut prices) or accommodate.
- If it fights, both firms make a lost (-4, -3). Therefore the dominant strategy for Firm B appears to be accommodate, leaving both firms with (1,1)
- However, firm B may make the calculation that it is worth making a temporary loss, in order to try and force the new firm out of business. Also, if firm B fights, it may deter other entrants.
- See also: Game theory – entry deterance

### Decision Tree

In this decision tree, player 1 can go high or low.

- If player 1 goes high, the dominant strategy for player 2 is to go high (3,5)
- If player 1 goes low, the dominant strategy for player 2 is to go high (10,4)
- Therefore, player 1 will choose low, because it knows that is the best choice.

#### Dominant strategy

A dominant strategy occurs when there is an optimal choice of strategy for each player no matter what the other does.

- If P2 chooses left P1 will choose UP
- If P2 chooses right P1 will choose UP
- Therefore UP is a dominant strategy for P1
- P2 will always choose right no matter what P1 does
- The unique equilibrium is (up, left). This is best for both.

### Nash Equilibrium

A Nash equilibrium occurs when the payoff to player one is the best given the other’s choice.

- In this case If P1 chooses down, P2 will choose right
- If P1 choose UP, P2 will choose right. But, if P2 choose right, P1 will want to choose down.
- The Nash equilibrium will be downright, (5,5) despite UP left being the optimal Pareto outcome.

### Collusion and game theory

- If firms are competitive and they set low price -they will both make £4m.
- If they collude and set high price, then they will both double their profits and make £8m.
- However, if during collusion, firm A undercuts the collusive price and sets a low price – it is able to sell more. In this case, firm A benefits from the best of both worlds. Prices are high because firm B is setting high price, but firm A is also selling large quantities because it is undercutting its rival. In this case, firm A makes £10m and firm B only makes £2m.
- Therefore, firm B is unlikely to keep prices high and the market reverts to both setting low prices.

The optimal outcome for the firms is to collude (high price, high price)

### Repeated Games and Game Theory

If games are repeated then there is the possibility of punishing people for cheating, this will provide an incentive for sticking to the Pareto optimal approach.

However, if they are repeated a finite number of times then there will be an incentive to cheat. If the game is played 10 times then the player will defect on the 10th round so why cooperate. So, therefore, you may as well defect on round 9 and so round 8 as well

If it is played an infinite number of times then it will be different. The best strategy then is to play tit for tat. If a player defects in one round you retaliate in the next round. In other words, you do whatever your opponent does and this is an incentive to enforce the cartel.

### Game Theory: A game of entry deterrence

If a new firm enters the market then the payoff will depend on whether the incumbent fights or accepts. If the incumbent fights they both get 0. If it does not fight then the incumbent gets 1 and the entrant gets 2. Therefore the equilibrium is for the new firm to enter and the incumbent to accept.

However, if the incumbent can give a credible threat that he will fight then he may be able to persuade the entrant to stay out. He could do this by investing in extra capacity, which would give him a bigger payoff in a price war. This would deter entry. So although the monopolist would never use this he would prevent entry.

Game theory and the kinked demand curve

### Game Theory can be used for pricing strategies

In oligopoly firms may be deciding whether to cut prices, increase prices or keep them static.

The kinked demand curve model suggests the most likely outcome is for price stability. This is because

- If firms increase the price, others don’t – Therefore demand falls significantly. (demand is elastic)
- If firms cut price, you would gain an increase in market share. Other firms don’t want to allow this. Therefore, they cut prices as well. Basically causing a price war where everybody loses out.

Therefore, in oligopoly, an important feature of firms decisions is the impact of interdependence. Decisions of one firm significantly impact on others.

### War vs Peace

In this case, the best outcome for both parties is Peace, Peace (a) – both get 100. However, if you think there might be war, it is better to strike first. Because if you strike first and start the war, you get 50, whereas the loser gets – 50.

If there is tension between the two, and you distrust the other country, then it could encourage a country to start a war and deviate from the best strategy for both.

**Related**

- Game theory – reducing the price of PlayStations