Exit barriers (or barriers to exit) are obstacles that stop or prevent the exit of a firm from a specific market. It is associated with firms that are incurring in some form of losses, but cannot exit the market as a result of exit barriers that would further increase their level of loss. In Michael Porter’s model of competitive analysis, barriers are a fundamental element to gauge the level of competition in a sector, and defines the market structure in that industry. He identified the following exit barriers:

Non-transferable assets: when a firm invests on specialised assets, which cannot be used in other industries, exiting the market implies losing those assets.

Fixed exit costs: such as indemnities paid to employees, cancelation costs in contracts with suppliers, etc.;

Synergies: when producing one good allows for lower costs in the production of other goods, exiting an industry may imply higher costs when producing that good.

Although barriers may have negative consequences for some firms, for others they will be positive. Where some firms loose other win, it can be regarded as a zero-sum game. For abnormal profits to occur in a market, barriers have to exist. Using Joe Bain’s definition, “barriers give firms the power to maintain in the long-term prices higher than average cost”.

In game theory, repeated games, also known as supergames, are those that play out over and over for a period of time, and therefore are usually represented using the extensive form. As opposed to one-shot games, repeated games introduce a new series of incentives: the possibility of cooperating means that we may decide to compromise in order to carry on receiving a payoff over time, knowing that if we do not uphold our end of the deal, our opponent may decide not to either. Our offer of cooperation or our threat to cease cooperation has to be credible in order for our opponent to uphold their end of the bargain. Working out whether credibility is merited simply involves working out what weighs more: the payoff we stand to gain if we break our pact at any given moment and gain an exceptional, one off payoff, or continued cooperation with lower payoffs which may or may not add up to more over a given time. Therefore, each player must consider their opponent’s possible punishment strategies.

This means that the strategy space is greater than in any regular simultaneous or sequential game. Each player will determine their strategies or moves taking into account all previous moves up until that moment. Also, since each player will take into account this information, they will play the game based on the behaviour of the opponent, and therefore must consider also possible changes in the behaviour of the latter when making choices.

Repeated games provide different payoffs at each repetition, depending on each player’s moves. Since these payoffs are given at different points in time, in order to analyse repeated games, we must compare each player’s discounted sum of payoffs, which for infinite repetitions and finite repetitions are calculated using the following formulae: 

Where:   -P: the discounted sum of payoffs;

-t: the number of the repetition being considered;

-n: the total number of repetitions for finite repeated games;

-p_t: the payoff at the repetition being considered;

-r: the discount rate.

Repeated Prisoner’s dilemma:

In the game known as the Prisoner’s dilemma, the Nash equilibrium is Confess-Confess (defect-defect). In order to see what equilibrium will be reached in a repeated game of the prisoner’s dilemma, we must analyse two cases: the game is repeated a finite number of times, and the game is repeated an infinite number of times.

When the prisoners know the number of repetitions, it’s interesting to operate a backwards induction to solve the game. Consider the strategies of each player when they realise the next round is going to be the last. They behave as if it was a one-shot game, thus the Nash equilibrium applies, and the equilibrium would be confess-confess, just like in the one-time game. Now consider the game before the last. Since each player knows in the next, final round they are going to confess, there’s no advantage to lie (cooperate with each other) on this round either. The same logic applies for prior moves. Therefore, confess-confess is the Nash equilibrium for all rounds.

The situation with an infinite number of repetitions is different, since there will be no last round, a backwards induction reasoning does not work here. At each round, both prisoners reckon there will be another round and therefore there are always benefits arising form the cooperate (lie) strategy. However, prisoners must take into account punishment strategies, in case the other player confesses in any round.

Collusion agreement games:

If we assume the game can be played ad infinitum, we can apply it in a collusion agreement game, where two firms collude, forming a cartel. Consider two firms (a duopoly) that may either behave as Cournot duopolists earning profits π_Cournot each, or collude and act as a cartel, earning π_Cartel each, which correspond to the profits of a monopoly divided into the number of firms colluding (two in our example).

In this case, we simply need to apply the formula for calculating an infinite sequence and a discount factor to compensate for the fact that the gains to be derived are over time (accounting for impatience, inflation, loss of interest, etc.): 

The left hand side represents the payoff derived from collusion, which can be held infinitely over time, with δ being the discount factor to bring future benefits forward to the present. For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel. The higher δ is, the higher the value assigned to future benefits and therefore the greater the chances of collusion. It is worth reminding here that fair competition is regulated in almost all countries, with cartels being banned, so most markets that lend themselves to reduced competition and price fixing are closely monitored.

Although this example is widely used in game theory and for the analysis of market structures, it can be easily seen that it does not represent a real situation. Let’s consider the same example: any of the colluding firms might deviate, in order to dump more output in the market at lower prices, in order to gain market share. This move will allow that firm to sell more products than the other firms, which directly contradicts Cournot’s premise that each duopolist will produce the same quantity. Therefore, considering a Stackelberg duopoly might seem more realistic. This would obviously change the analysis and outcome of the game.

olk theorems are used in Economics specially in the field of game theory and specifically to repeated games. This theorem is said to be satisfactorily fulfilled when the equilibrium outcome in a game that is repeated an infinity number of times, is the same as the feasible and strongly individually rational outcome in the one-shot game. The outcome is said to be in equilibrium, because any attempt of trying to increase a player’s individual outcome will imply a decrease in at least another player’s individual outcome.

The origin of this theorem is unknown but it appeared in the late fifties, and became quickly known and spread within the game theory academia, and thus the reason why it became known as the folk theorem.  James W. Friedman in his article “A Non-cooperative Equilibrium for Supergames”, 1971, was the first to write and publish a paper in which he thoroughly deals with the theorem.

The main issue that this theorem explains, is the strong relationship between repeated and cooperative games, revealing the enforcement mechanism characteristic of repetition. One of the first applications of the theorem was to collude when dealing with a Cournot duopoly:
 

The left hand side represents the payoff derived from collusion, which can be held infinitely over time, with δ being the discount factor to bring future benefits forward to the present. For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel. The higher δ is, the higher the value assigned to future benefits and therefore the greater the chances of collusion. It is worth reminding, that fair competition is regulated in almost all countries, with cartelsbeing banned, and so most markets that lend themselves to reduced competition and price fixing, are closely monitored.

Collusion makes allusion to the cooperation between different firms. This cooperation leads to a restrain of market competition, in any of its forms, which translates into higher profits for the firms in detriment of consumer’s welfare.  A cartel is an example of firms belonging to the same industry structure which collude to some degree in setting prices and/or output levels. Agreements which have as their object or effect the prevention, restriction or distortion of perfect competition are prohibited. Such agreements include, but are not restricted to, activities such as:

-fixing purchase or selling prices or any other trading condition, directly or indirectly;

-controlling or limiting production levels, markets, technological advances or investments;

-sharing markets or resources supplies.

Legislation in different countries may consider different scenarios and penalties for such agreements, but the main idea is clear: firms behaviour shall not affect the correct functioning of market forces.

A clear example is to consider an industry where there are only two firms (duopoly). Both firms will set their levels of output and prices with the objective of maximizing their joint profits. Many strategies can be used in order to maximize profits which would lead to a multiple Nash equilibria solution. As seen in the figurebelow, collusion maximises aggregate profits for both firms, since the isoprofit curves are tangent. It’s a better equilibrium than the one in Cournot duopolies or Stackelberg duopolies.

However, cartels are not stable. There will always be incentives for each firm to trick the other, and change their output and/or price level in such a way, that they’ll increase their own profit in detriment of the other’s. To avoid this practice, any deviation by any party should be instantly punished; this is known as a trigger strategy. James W. Friedman demonstrated in his paper “A Non-cooperative Equilibrium for Supergames”, 1971, that in this context of infinite interactions, it is possible that collusion occurs due to this punishment strategy. That is, the cartel may endure as long as punishment strategies are so devastating that the benefits derived form deviation would end up being smaller than the benefits of keep colluding. J. W. Fridman put this idea in what is known in game theory as Folk theorem: 

The sustainability of the equation will depend mainly in two factors: the credibility of the threat of punishment, and the discount factor. The former is easily understood as a credible threat will ensure no deviations are made, and the latter is related with how much does each party value the profits obtained from the results of following a collusive strategy, compared to the possible profits of changing their strategy.

Factors that guarantee collusion stability:

There are number of factors that affect this collusive equilibrium, such as:

-Number of firms in the market: the higher the degree of concentration in a market the higher the incentives to collude. Firms in highly concentrated markets will tend to collude since all the profits will be distributed amongst fewer firms.

-Multimarket contact: if firms compete in more than one market, the collusive agreement will be more stable. Firms that compete with other firms over many markets can establish trigger strategies that can be applied in all these markets, which will create a more devastating punishment strategy.

-Market transparency: the more transparent a market is, the easier it is to ensure that every firm is following the same strategy and is not deviating from the deal. Collusion will be more difficult in industries where it is harder to detect changes in firm’s prices or output.

-Asymmetry between firms: the bigger the asymmetry between firms, the harder it is for collusion to take place. If firms have different cost structures, the one with the lowest costs will be incentivised to lower its prices, and thus cause the other firm to have to exit the market.

Collusion agreement games:

In game theory, collusion agreements can be described using the extensive form, as depicted in the adjacent game tree. In this case, two firms share the market, already colluding and maintaining high prices. Each firm can decide to stop colluding and start a price war, in order to increase their market share, even force the other to quit the market. Firm 1 can either keep colluding with firm 2, or start a price war. If firm 1 decides to keep colluding, firm 2 will need to make a decision. If they both agree to collude, they will get 5,5. However, if one of them decides to start a price war, the set of payoffs will be either 4,3 or 3,4, depending on which one starts the war (and therefore acquires a greater market share). It’s easy to see that collude-collude is both the Nash equilibrium and a Pareto optimum situation. This result may change when considering repeated games, as seen before.

Entry barriers (or barriers to entry) are obstacles that stop or prevent the entrance of a firm in a specific market. It is associated with the situation in which a firm wants to enter a market due to high profits or increasing demand but cannot do so because of these barriers. In Michael Porter’s model of competitive analysis, barriers are a fundamental element to gauge the level of competition in a sector, and relates to the market structure. Here are some of these entry barriers:

Economies of scale, economies of scope: any of these could be necessary to enter an industry, since fewer costs can mean the possibility to remain in that industry;

Product differentiation: if incumbent firms in the industry have from customers a certain degree of loyalty towards their products, differentiation may be its origin;

Minimum capital requirements: in some industries, high investments may be needed in order to be able to produce;

Complicated change of supplier: if our potential customers find it hard to change suppliers, they might not be willing to change;

Access to distribution channels: of special interest when considering consumer goods, it means that getting the distribution channels to distribute your product may prove hard;

Costa advantages other than economies of scale: such as technology, know-howor simply because of the learning curve;

Government regulation: requirement of permits or licenses such as construction permits or taxi licenses;

Expected reaction from incumbents: if price wars or predatory pricing is expected, this could act as a barrier to entry.

Although barriers may have negative consequences for some firms, for others they will be positive. Where some firms loose other win, it can be regarded as a zero-sum game. For abnormal profits to occur in a market, barriers have to exist. Using Joe Bain’s definition, “barriers give firms the power to maintain in the long-term prices higher than average cost”.

The figure below shows the relation between entry barriers and market structure:

The Edgeworth duopoly model, also known as Edgeworth solution, was developed by Francis Y. Edgeworth in his work “The Pure Theory of Monopoly”, 1897. It is a duopoly model similar to the duopoly model developed by Joseph Bertrand, in which two firms producing the same good compete in terms of prices. Edgeworth’s model presents a slight modification as it also includes constraints in the production capacity of the firms. In this market structure, firms have two potential options, to collude or not.

As shown in the adjacent figure, when firms choose to collude they will split and share the market and the production of the good. Firm₁ will produce from O to F and firm₂ from O to G, in this way the supply is limited and prices will be set at p. Revenues of each firm correspond to the rectangle above FO and OG, and each firm would enjoy an equal share. Note that d₁ and d₂ are parts of total demand, each part being supplied by one of the firms.

Collusion is not always possible as firms have incentives to break cooperation in their search for higher profits. Collusion is also considered an illegal business practice in many countries. Eventually one of the firms will decide to lower their prices and increase production in order to gain market share from the other competitor.  Consequentially the other firms will do the same. This process will escalade up to the point in which the maximum production of both firms is achieved. When this point is reached (OD for firm₁ and OE for firms₂), price will not be reduced any further and will remain at p’, as the increase in demand that follows price reduction will not be satisfied with a larger amount of production. On the contrary, prices will start to rise little by little so firms will be able once again to increase their profits. Overtime this process will be repeated and prices will oscillate from p to p’.

The following figure shows how all this translates into market demand (D), and how quantities sold will oscillate from Q and Q’.

Overall the Edgeworth’s solution is a more realistic one than Bertrand’s and it answers Bertrand’s paradox. However, it does not give a definitive solution.

In some cases, competition in terms of price changes seems more logical than quantity competition, especially in the short run. Besides, one of the assumptions of Cournot’s duopoly model is that firms supply a homogeneous product. Considering this, Bertrand proposed an alternative to Cournot. Considering Bertrand’s model from a game theory perspective, it can be analysed as a simultaneous game where the strategic choice is on prices, rather than quantities.

In this model, consumers will buy from the firm that offers the lowest price, so we can easily have the intuition that the Nash equilibrium is going to be the two firms setting the same price.

Assuming equal and constant cost functions, the demand for each firm is as follows: 

Bertrand’s equilibrium occurs when P₁=P₂=MC, being MC the marginal cost, yielding the same result as perfect competition.

The logic is simple: if the price set by both firms is the same but the marginal cost is lower, there will be an incentive for both firms to lower their prices and seize the market. Therefore, the only equilibrium in which none of the firms will be willing to deviate is when price equals marginal cost. 

The result of the model creates a paradox, known as Bertrand’s paradox: in a case of imperfect competition (here, a duopoly), where there is a strong incentive to collude, we end up with the same outcome as in perfect competition. The equilibrium does not hold with asymmetric cost functions since the firm with the lowest marginal cost would seize the entire market and become a monopoly. A great analysis of this paradox, known as Edgeworth duopoly model or Bertrand-Edgeworth duopoly, was developed by Francis Y. Edgeworth in his paper “The Pure Theory of Monopoly”, 1897.