Question

Tourists in Bintan island, a 45-minute ferry ride from Singapore, are demanding the latest espresso drinks from a familiar brand but their small number means demand can only sustain one new coffee shop. You and another popular chain, Starbucks, are considering making the considerable investment necessary to set up a shop in Bintan. Analyse the interaction between the two firms using game theory. Present a payoff matrix to model the situation and analyse it for Nash equilibrium. What can you do to make your best outcome more likely?

Answer 1

Game theory is widely regarded as having its origins in the mid-nineteenth century with the publication in 1838 of Augustin Cournot’s Researches into the Mathematical Principles of the Theory of Wealth, in which he attempted to explain the underlying rules governing the behaviour of duopolists. However, it was with the publication in 1944 of John von Neumann and Oskar Morgenstern’s The Theory of Games and Economic Behaviour that the modern principles of game theory were formulated. Game theory has been widely applied to the behaviour of producers with a few or only one competitor

1. Rules, which govern conduct of the players
2. Pay-offs, such as win, lose or draw
3. Strategies, which influence the decision making process.

In applying game theory to the behaviour of firms we can suggest that firms face a number of strategic choices which govern their ability to achieve a desired pay-off, including:

Decisions on price and output, such as whether to:

• Raise
• Lower
• Hold

Decisions on products, such as whether to:

• Keep existing products
• Develop new ones

Decisions on promoting products, such as whether to:

• Spend more on advertising
• Spend less
• Keep spending constant

Firms could derive a range of possible pay-offs from their strategy choices, including:

• More profits for shareholders
• Greater market share
• Improved chances of survival
• Getting rid of a rival

A maximax strategy is one where the player attempts to earn the maximum possible benefit available. This means they will prefer the alternative which includes the chance of achieving the best possible outcome – even if a highly unfavourable outcome is possible.

This strategy, often referred to as the best of the best is often seen as ‘naive’ and overly optimistic strategy, in that it assumes a highly favourable environment for decision making.

A maximin strategy is where a player chooses the best of the worst pay-off. This is commonly chosen when a player cannot rely on the other party to keep any agreement that has been made – for example, to deny. In the Prisoner’s Dilemma, the worst pay-off to Robin from confessing is to get 3 years (with Tom confessing), and the worst pay-off from denying is 10 years (with Tom confessing) – therefore the best of the worst is to confess.

In this case, both the maximin and maximax strategies would be to confess. When this occurs, it is said to be the dominant strategy.

A dominant strategy is the best outcome irrespective of what the other player chooses, in this case it is for each player to confess – both the optimistic maximax and pessimistic maximin lead to the same decision being taken.

In general, game theory suggests that firms are unlikely to trust each other, even if they collude and come to an agreement such as raising price together.

Consider the hypothetical example of two Airlines and return ticket prices to New York.

In this case, for both Airlines, the aggressive maximax strategy is £140m from a low price and £120m from a high price, so a low price gives the maximax pay-off.

In terms of the pessimistic maximin strategy, the worst outcome from a low price is £100m, and from a high price is £70m – hence a low price provides the best of the worst outcomes.

Again, lowering price is the dominant strategy, and the only way to increase the pay-off would be to collude and increase price together. Of course, this requires an agreement, and collusion, and this creates two further risks – one of the airlines reneges on the agreement and ‘rats’, and the competition authorities investigate the airlines, and impose a penalty.

Nash equilibrium, named after Nobel winning economist, John Nash, is a solution to a game involving two or more players who want the best outcome for themselves and must take the actions of others into account. When Nash equilibrium is reached, players cannot improve their payoff by independently changing their strategy. This means that it is the best strategy assuming the other has chosen a strategy and will not change it. For example, in the Prisoner’s Dilemma game, confessing is a Nash equilibrium because it is the best outcome, taking into account the likely actions of others.