Reaction Functions and ISO-Profit Curves

Competition can be described as a game of non co-operation amongst the firms. Game theory provides an important tool of analysis in understanding the behaviour of modern firms as players in a game of non co-operation. The maxim that trying to do better than the cartel outcome (2,2) but always doing worse at (1,1) holds true as each player will attempt to do better than the agreed or arrived at collective equilibrium. This is also referred to as the Prisoner’s dilemma.

Here we represent the possible outcomes to demonstrate the maxim with one player seeking to do better than (2,2) by obtaining a 3 but ends up at (1,1). The player can only obtain 3 if the other player receives 0. In other words, one player opts to go for a low price but only if a second player keeps its price high - that is the only way to secure a 3. But the second player soon realises that 0 is an outcome, so that player also reduces its price to low and both players end up at (1,1).

The only solution is communication and this is illegal in the real world due to the antitrust legislation on cartels and price fixing. In other words, one player, a price leader, initiates an agreement to remain at the (2,2) outcome. In many markets, a fact finder would observe constant or fixed prices. However the mere adherence to a fixed price is not sufficient evidence of belonging to a cartel as there must be some evidence of a rule or mechanism to ensure that the (2,2) outcome obtains across all periods and there is no incentive to cheat because of (say) a punishment strategy.

If both players were to communicate over a four period game then the total payoffs would amount to 2+2+2+2 = 8. If one player deviates from the agreement and cheats by charging a low price in the second period, that player obtains 2+3 = 5. However, the other player observes the cheating behaviour and reduces its price to low and does not move in order to punish the first player who now obtains 2+3+1+1 = 7 with a realisation of 1+1+1+1 = 4 for all periods unless there is an agreement not to cheat. But this is difficult to maintain in the real world unless a credible punishment mechanism can be put in place by one of the players. Competition enforcement agencies now rely on whistle-blower legislation to entice a cartel member to come forwards and reveal the cartel mechanism. Alternatively, with an incentive for a player to do better, any cartel is inherently unstable.

[2.2]

we can both have the payoff 2

]]> [0.3]

you want this payoff 3 instead

]]> [3.0]

]]> [1.1]

we both end up with a 1 payoff

]]>

The histogram illustrates the Prisoner’s Dilemma again.

The two firms are faced with a set of choices: Choice 1 would secure both of them a profit of (500,500). However, one of the firms, the Green firm, believes it can do better than 500 and moves to secure 800. However, as Choice 2 illustrates this leads to a lower profit for the red firm at 150. Choice 3 occurs when the red firm believes it can do better than 500. Both firms believe that they can do better but it is contingent on the other firm not responding. For example, once the green firm secures 800, the red firm will observe this advantage – it may have happened due to the green firm securing a price differentiation advantage over the red firm’s product. Hence the red firm may follow and also reduce price in order to gain additional profit. But by unilaterally trying to do better than (500,500) both firms end up worse off at Choice 4 at (300,300). Choice 4 is the best that each can do given the reaction of the other – it is a Nash equilibrium.

Key Points

Competition can be described as a game of non co-operation amongst the firms. Game theory provides an important tool of analysis in understanding the behaviour of modern firms as players in a game of non co-operation. The maxim that trying to do better than the cartel outcome (2,2) but always doing worse at (1,1) holds true as each player will attempt to do better than the agreed or arrived at collective equilibrium. This is also referred to as the Prisoner’s dilemma.

Here we represent the possible outcomes to demonstrate the maxim with one player seeking to do better than (2,2) by obtaining a 3 but ends up at (1,1). The player can only obtain 3 if the other player receives 0. In other words, one player opts to go for a low price but only if a second player keeps its price high - that is the only way to secure a 3. But the second player soon realises that 0 is an outcome, so that player also reduces its price to low and both players end up at (1,1).

The only solution is communication and this is illegal in the real world due to the antitrust legislation on cartels and price fixing. In other words, one player, a price leader, initiates an agreement to remain at the (2,2) outcome. In many markets, a fact finder would observe constant or fixed prices. However the mere adherence to a fixed price is not sufficient evidence of belonging to a cartel as there must be some evidence of a rule or mechanism to ensure that the (2,2) outcome obtains across all periods and there is no incentive to cheat because of (say) a punishment strategy.

If both players were to communicate over a four period game then the total payoffs would amount to 2+2+2+2 = 8. If one player deviates from the agreement and cheats by charging a low price in the second period, that player obtains 2+3 = 5. However, the other player observes the cheating behaviour and reduces its price to low and does not move in order to punish the first player who now obtains 2+3+1+1 = 7 with a realisation of 1+1+1+1 = 4 for all periods unless there is an agreement not to cheat. But this is difficult to maintain in the real world unless a credible punishment mechanism can be put in place by one of the players. Competition enforcement agencies now rely on whistle-blower legislation to entice a cartel member to come forwards and reveal the cartel mechanism. Alternatively, with an incentive for a player to do better, any cartel is inherently unstable.