How can game theory be used by competing firms in competitive markets to maximise profits

How can game theory be used by competing firms in competitive markets to maximise profits?
Introduction and defining the terms used in game theory.

Game theory is defined by as “the formal study of decision making where several players must make choices that potentially affect the interest of other players”, and is used in a wide variety of fields of study; consisting of economics, sport, politics, law and even biology. The game itself was invented by John von Neumann and Oskar Morgenstern in 1944 when they co published ‘Theory of Games and economic behaviour’, which explained the mathematical aspects behind the game and how it is played.
To first understand game theory, there are certain terms that must be defined. Players are described as “Any participant in a game who (i) has a nontrivial set of strategies (more than one) and (ii) selects among the strategies based on payoffs. If a player is a non-strategic, selecting strategies randomly, the player is termed a nature player”. This implies that every participant in the game has two or more options, i.e. to set the price of a product at £1, or £1.10. Generally, in open markets we assume firms have more than two decisions, as they have the ability to set their price at any level, thus I will be using four strategies from here on. This is based on the basic economic theory that any large increase, or decrease in price will result in either a fall in demand, and thus revenue, or a large increase in supply but widely reduced profit margins.

A strategy, as defined by Investopedia, is “A complete plan of action a player will take given the set of circumstances that might arise within the game” and dictates how the game plays out. Various strategies, when applied to firms, usually consist of raising, or lowering the price of the product. This will in turn lead to the payoff, or the outcome of using that strategy.

The prisoner’s dilemma
Perhaps the most famous example of game theory is the prisoner’s dilemma, which was created in 1950 by Merrill Flood and Melvin Dresher, which was then altered and renamed by Albert W. Tucker.

The game works as such, two prisoners can either confess or remain silent, these are the two strategies. The payoffs are the number of year’s imprisonment they receive and the players are the prisoners.
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From here we see the concept of the Nash equilibrium, which is defined as where players ‘have no incentive to change their strategy. Having reached a Nash Equilibrium a player will be worse off by changing their strategy’. In the case of the Prisoners Dilemma the optical scenario is both prisoners denying, however the rational solution is that they both confess, as one may undercut the other (something I will look into further later on). Thus we reach a Nash equilibrium, where no player has an incentive to deviate from his strategy as they receive no benefit- in fact they run the risk of being imprisoned for 20 years as opposed to 5 years or 0 years.

Playing a game
From this we can begin the formation of a table or tree diagram to illustrate a Game. The rules of this game are as follows:
Neither players can deviate from the prices chosen
If player 1 choses to set the price of the product at £1 when player 2 sets the price of the product at £1 then the payoff is £100 million for player 1
If player 1 choses to set the price of the product at £1 when player 2 sets the price of the product at £2 then the payoff is £140 million for player 1
If player 1 choses to set the price of the product at £2 when player 2 sets the price of the product at £1 then the payoff is £70 million for player 1
If player 1 choses to set the price of the product at £2 when player 2 sets the price of the product at £2 then the payoff is £115 million for player 1
The above rules apply as inverse for player 2
Player 2 £1 £2
Player 1 £1 £100 million £140 million
£2 £70 million £115 million
The table above shows Player 1’s payoff.

Player 2 £1 £2
Player 1 £1 £100 million £70 million
£2 £140 million £115 million
The table above show’s Player 2’s payoff.

Here we can see two separate tables which show the revenue for both firms separately, however to make things even simpler, we can superimpose the two tables to form one, simple matrix that denotes the outcomes for both firms.

Player 2 £1 £2
Player 1 £1 £100m, £100m £140m, £70m
£2 £70m, £140m £115m, £115m
This final table has the same basic structure, however shows the payoffs of Player 1 and Player 2 respectively, i.e. if player 1 choses £1 and player 2 choses £2, then as shown by the matrix Player 1’s revenue is £140m and Player 2’s revenue is £70 million. This form of matrix is known as an ‘outcome matrix’ and tells us everything in the game.

Assumptions about the motives of firms and competition
In the twentieth century, a “firm”, commonly known as a “business” is an ‘organisation that employs productive resources to obtain products and/or services which are offered in the market with the aim of making a profit’. Already we assume that the first aim of a firm is profit maximisation, and thus they will seek to earn as much revenue as possible, provided that the cost to manufacture the good remains the same (known as ceteris paribus in economic terms, ‘all other things being unchanged’). This means that from the strategies available, we can establish a firm is most likely to choose one that has the least risk of a deduction in revenue.

There are, however, instances where firms do not carry out the objective of profit maximisation, for example Charities, non-profit organisations and firms such as Amazon that are looking to expand their market share and increase revenue as opposed to profit. Game theory can be applied to these firms, however in this instance we will not be doing so as we are looking at how firms can maximise profits in markets through game theory, and not any other situation.

We also must assume that consumers, who by definition is ‘someone who purchases and uses products and services for personal use’, are utility maximiser. By this we mean that they seek to maximise the amount of satisfaction from the utility they gain, from the least amount of expenditure.

Despite this, there are also certain situations in markets where profit motivated firms cannot have game theory applied to them, one of these being a situation of perfect competition. This is a situation in the market where there are (i) many buyers and sellers, (ii) homogenous, or, identical goods being sold by firms to consumers, (iii) perfect information- all consumers know when there is a price change and knowledge is available to consumers free of charge, (iv) no barriers of entry or exit into the firm- any firm can enter the market resulting in large numbers of alternatives for consumers to choose from, (v) ‘there is no need for government regulation except to make markets more competitive’ . As a result of this, if one firm were to increase the price of a good, this would result in a serious reduction in revenue and thus profit, as consumers would switch to one of many alternate firms in the market currently selling at a lower price. This can be demonstrated via a diagram of ‘perfectly elastic demand’. The price determined for the good comes from the intersection of the demand and supply curve on the diagram on the left.

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The demand curve above is not only the average revenue curve but is also the marginal revenue curve. This means that any price that is set above the curve results in an instant loss in revenue and profit. A reduction in the price however will also lead to a reduction in revenue, this is because the maximum profit that can be made by a firm lies on the intersection of the Marginal cost and Marginal revenue. Thus we establish in a situation of perfect competition firms have no choice but to remain at the same price.

Monopolies
A similar situation to perfect competition, however a monopoly is defined as ‘where one producer controls supply of a good or service, and where the entry of new firms is prevented or highly restricted’. This indicates that a market is controlled by one firm, and so firms do not need to make decisions about pricing strategies based on the strategies of other firms. By this game theory cannot be applied as by definition it requires two or more players.

How game theory can be used by firms
The first instance we will use is a Duopoly, which as defined as ‘a situation in which only two suppliers dominate the market for a commodity or service’. This means that only two firms have a majority of the market share, and so they are only each other’s competition, and thus a game can begin to be formed from this. An example of a duopoly is Seagate and western Digital’s combined 84% market share of the hard disk drive market (worth $23.9bn)
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According to the quarterly earnings statistics of Western Digital, in the financial year of 2014 they generated an average of $3.7825bn per quarter, giving total revenue to be $15.130bn. Seagate, from their published quarterly earnings had an average quarterly revenue of $3.431bn, giving an annual revenue of $13.724bn. The average price per Gb of a hard drive in 2014 was $35 per Terabyte, however due to the Thailand flood in 2012 prices were decreasing by an average of 32% per annum. However, prices were raised instead, even though demand nor supply had changed, thus we can establish that an element of price fixing was involved, and the same situation form the prisoner’s dilemma has occurred.

Where prices should have been $20.23 per terabyte, the duopolistic firms instead earned a combined revenue of $28.854bn. Seagate sold 224million units of HDD in 2014 , meaning that as the average space per hard drive was 1Tb in 2014, $7.84bn came from selling HDD’s at $35 that year (224,000,000×35=7.8bn), Western Digital sold 249million units in 2014, so given the same average space per drive their revenue from HDD’s was $8.715bn- from here we can begin to form a game likely used at the time.

Western Digital $20.23 $35
Seagate $20.23 $35 $7.84bn,$8.715bn
Now, as the price of HDD’s should have been $20.23 (referenced earlier), then Seagate’s revenue should have been 224,000,000×20.23= $4.53bn, and Western Digital’s should have been 249,000,000×20.23= $5.03bn, which we can enter into our matrix.

Western Digital $20.23 $35
Seagate $20.23 $4.53bn/$5.03bn $35 $7.84bn,$8.715bn
Although there are no exact figures for what the revenues would have been, however we can estimate that at least 35% of business would have been lost to the other competing firm, and so our table would look something like this:
Western Digital $20.23 $35
Seagate $20.23 $4.53bn/$5.03bn $6.294bn, $5.665bn
$35 $5.096, $6.623bn $7.84bn,$8.715bn
In this case for Game theory, the dominant strategy would be to set the price at $35, as even if firms lose market share by raising the price when one keeps it lower, both firms see large increases in revenue. The dominated strategy in this case is setting the price at $20.23, and in 2007 Professor Ben Polak from Yale university once said “don’t play a strictly dominated strategy”, his reasoning: “If I instead play the strategy that dominates it, I do better in every case”.

A dominant strategy, as seen above, is defined as ‘regardless of what any other players do, the strategy earns a larger payoff than any other.’ This is clearly the case here where In every case charging $35 per Tb earns a much higher profit, and, ceteris paribus, as the cost to manufacture and develop the hard drives remain the same, then the duo in turn receive much higher quarterly profits.

An alternate example where increasing the price is not the dominant strategy is when price undercutting may be used. Price undercutting, defined as ‘when a firm sells a good or service at a price below cost (or very cheaply) with the intention of forcing rival firms out of business.’ This can be used in price wars between supermarkets, an example of this being supermarkets in the UK such as Aldi, Sainsbury’s, Tesco, Morrisons and Asda. Aldi recently have lost short term profits via the use of predatory pricing, however their intention is to see long term increases in profit once their market share has increased.

Bibliography
BIBLIOGRAPHY Binmore, K. (1991). Fun and games.
Stegnel, B. v. (2001). Game Theory. London: London School of Economics.

Smith, Peter. (2015). Economics for A level year 1 and AS. London, Hodder Education