Folk 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.

In game theory, a subgame is a subset of any game that includes an initial node (which has to be independent from any information set) and all its successor nodes. It’s quite easy to understand how subgames work using the extensive form when describing the game. In the following game tree there are six separate subgames other than the game itself, two of them containing two subgames each.

A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. Consider the following game: player 1 has to decide between going up or down (U/D), while player 2 has to decide between going left or right (L/R). In this case, we can represent this game using the strategic form by laying down all the possible strategies for player 2:

-go Right if player 1 goes Up, go Left otherwise;

-go Left if player 1 goes Up, go Right otherwise;

-go Right no matter what;

-go Left no matter what.

We can see how this game is described using the extensive form (game tree on the left) and using the strategic form (game matrix on the left). Since this is a sequential game, we must describe all possible outcomes depending on player 2 decisions, as seen in the game matrix. 

If we look for the equilibrium of this game, considered as a whole, we find that Up-Left is a Nash equilibrium (red). However, it’s not a perfect equilibrium. In order to find the subgame-perfect equilibrium, we must do a backwards induction, starting at the last move of the game, then proceed to the second to last move, and so on. In this particular case, we know that player 2 will choose Left if player 1 goes Up, and Right if player 1 goes Down, since these are the moves that maximise his utility. Because there is complete information (and therefore each player’s payoffs are known), player 1 knows these choices in advance, and will therefore choose to go Down, because the payoff will be greater. Therefore, Down-Right is the perfect subgame equilibrium (green).

In sequential games, a series of decisions are made, the outcome of each of which affects successive possibilities. In game theory, the analysis of sequential games is of great interest because they usually model reality better than simultaneous games: producers will usually observe demand before deciding how much output to produce, duopolists will observe each other’s decisions before dumping more goods in the market, etc.  For example, let’s examine the decision a company faces when trying to break a market which is currently a monopoly. Sequential games are represented through decision trees, with successive nodes at each decision point:

The game represented in this decision tree shows firm 1 choosing whether to compete in a monopolistic market or not. As there are no existing competitors in the market, should they choose not to enter, its payoff will be zero and the existing monopoly will have a payoff of two. However, if firm 1 chooses to enter we reach node two, where the second decision must be made by the challenged player (the monopoly), whether to accommodate the new competitor or fight back to try and block its entry. In this particular example, if they choose to fight, both players will have a payoff of -1, whereas if the monopoly decides to accommodate the new player, our monopoly becomes a duopoly and the resulting drop in prices creates larger market demand, meaning that the existing company will still receive a payoff of one and the new player a payoff of two.

So, what can we assume will happen? We assume that both players dispose of complete and perfect information. They have both carried out a thorough market study and know the consequences of each outcome. In that case, what they can do is work backwards to know what the competition will do. If firm 1 enters the market, player 2 has no reason to fight back, as its payoff will suffer by two points. This means that player 2 knows what the outcome will be if they choose to enter, so all they have to do is choose between a payoff of two if the new firm enters the market or zero if it doesn’t. In this case, it is pretty clear what the outcome will be. The game would change again, of course, if successive nodes were introduced, where fighting could again flip the tables.

Cournot duopoly, also called Cournot competition, is a model of imperfect competition in which two firms with identical cost functions compete with homogeneous products in a static setting. It was developed by Antoine A. Cournot in his “Researches Into the Mathematical principles of the Theory of Wealth”, 1838. Cournot’s duopoly represented the creation of the study of oligopolies, more particularly duopolies, and expanded the analysis of market structures which, until then, had concentrated on the extremes: perfect competition and monopolies.

Cournot really invented the concept of game theory almost 100 years before John Nash, when he looked at the case of how businesses might behave in a duopoly. There are two firms operating in a limited market. Market production is: P(Q)=a-bQ, where Q=q₁+q₂ for two firms. Both companies will receive profits derived from a simultaneous decision made by both on how much to produce, and also based on their cost functions: TC_i=C-q_i.

So, algebraically:

In order to maximise, the first order condition will be:

And, if q_i=q_j, then both equal:

Therefore, the reaction functions (blue lines), where the key variable is the quantity set by the other firm, will take the following form:

What all this explains is a very basic principle. Both companies are vying for maximum benefits. These benefits are derived from both maximum sales volume (a larger share of the market) and higher prices (higher profitability). The problem stems from the fact that increasing profitability through higher prices can damage revenue by losing market share. What Cournot’s approach does is maximise both market share and profitability by defining optimum prices. This price will be the same for both companies, as otherwise the one with the lower price will obtain full market share, which makes this a Nash equilibrium, also known for this model the Cournot-Nash equilibrium.

If we consider isoprofit curves (those which show the combinations of quantities that will render the same profit to the firm, red curves) we can see that the equilibrium of the game is not Pareto efficient, since isoprofit curves are not tangent. The outcome is below that of perfect competition and therefore is not socially optimal, but it is better than the monopoly outcome.

Extending the model to more than two firms, we can observe that the equilibrium of the game gets closer to the perfect competition outcome as the number of firms increases, decreasing market concentration.

Comparison with Stackelberg duopolies:

-Cournot’s model is a simultaneous game, Stackelberg’s is a sequential game;

-In Cournot duopolies quantity sold is the same for both firms, while in Stackelberg duopolies, the quantity sold by the leader is greater than the quantity sold by the follower;

-When comparing each firm’s output and prices, we have:

Leader: q^S₁ > q^C₁ and π^S₁ > π^C₁

Follower: q^S₂ < q^C₂ and π^S₂ < π^C₂

-With regard to total output and prices we have the following:

Q_M < Q_C < Q_S < Q_PC

P_M > P_C > P_S > P_PC = MC

with:

Q_C: total Cournot output
Q_S: total Stackelberg output
Q_PC: total perfect competition output
Q_M: total monopoly output
P_C: Cournot price
P_S: Stackelberg price
P_PC: perfect competition price
P_M: monopoly price
MC: marginal cost

Mixed strategies need to be analysed in game theory when there are many possible equilibria, which is especially the case for coordination games. The battle of the sexes is a common example of a coordination game where two Nash equilibria appear (underlined in red), meaning that no real equilibrium can be reached.

In the battle of the sexes, a couple argues over what to do over the weekend. Both know that they want to spend the weekend together, but they cannot agree over what to do. The woman prefers to go shopping for a new pair of shoes, whereas the man wants to go a boxing match. The game matrix is therefore as follows:

In this case, knowing your opponent’s strategy will not help you decide on your own course of action, and there is a chance an equilibrium may not be reached. The way to solve this dilemma is through the use of mixed strategies, in which we look at the probability of our opponent choosing one or the other strategy and balance our pay off against it.

Let’s suppose that the woman is likely to choose boxing with probability q and shopping with probability (1-q). Likewise, the man is likely to choose boxing with a probability of r and shopping with a probability of (1-r). In that case, our outcomes are as follows:

• Boxing-Boxing: qr
• Shopping-boxing: (1-r) q
• Boxing-shopping: r (1-q)
• Shopping-shopping: (1-q) (1-r)

The man’s chances of going to a boxing match, his expected utility, will be 2r (payoff*probability) and, of shopping, 1-r (because his utility from shopping is 1), therefore r= 1/3.

Analogously, for the woman, q= 2/3. Now she must balance what q (the man’s chances of valuing his own happiness over hers) really is. If r>1/3, they’ll go to a boxing match. If r=1/3, either could happen, and if r<1/3, the woman will get her own way and they’ll go shopping. She must balance this carefully, because if she makes a mistake in valuing his probability (likewise for the man) then, as this is still a simultaneous game and there are no second chances, they could end up spending the weekend apart, which would mean less utility for both.

In the battle of the sexes, a couple argues over what to do over the weekend. Both know that they want to spend the weekend together, but they cannot agree over what to do. The man prefers to go watch a boxing match, whereas the woman wants to go shopping. This is a classical example of a coordination game, analysed in game theory for its applications in many fields, such as business management or military operations.

Since the couple wants to spend time together, if they go separate ways, they will receive no utility (set of payoffs will be 0,0). If they go either shopping or to a boxing match, both will receive some utility from the fact that they’re together, but one of them will actually enjoy the activity. The description of this game in strategic form is therefore as follows:

In this case, knowing your opponent’s strategy will not help you decide on your own course of action, and there is a chance an equilibrium may not be reached. This can be easily seen by looking for a dominant strategy, eliminating all dominated strategies. However, there will be two dominant strategies, two Nash equilibria (underlined in red). The way to solve this dilemma is through the use of mixed strategies, in which we look at the probability of our opponent choosing one or the other strategy and balance our pay off against it.

Dominant strategies are considered as better than other strategies, no matter what other players might do. In game theory, there are two kinds of strategic dominance:

-a strictly dominant strategy is that strategy that always provides greater utility to a the player, no matter what the other player’s strategy is;

-a weakly dominant strategy is that strategy that provides at least the same utility for all the other player’s strategies, and strictly greater for some strategy.

A dominant strategy equilibrium is reached when each player chooses their own dominant strategy. In the prisoner’s dilemma, the dominant strategy for both players is to confess, which means that confess-confess is the dominant strategy equilibrium (underlined in red), even if this equilibrium is not a Pareto optimal equilibrium (underlined in green).

It must be noted that any dominant strategy equilibrium is always a Nash equilibrium. However, not all Nash equilibria are dominant strategy equilibria.

The elimination of dominated strategies is commonly used to simplify the analysis of any game. The way to proceed is to eliminate for each player every strategy that seems ‘unreasonable’, which will greatly reduce the number of equilibria. This method is quite easy to use when only strictly dominated strategies are in place, but the elimination of weakly dominated strategies can turn problematic, ending up with a game that does not resembles the original one from a strategic point of view.

A good example of elimination of dominated strategy is the analysis of the Battle of the Bismarck Sea. In this game, as depicted in the adjacent game matrix, Kenney has no dominant strategy (the sum of the payoffs of the first strategy equals the sum of the second strategy), but the Japanese do have a weakly dominating strategy, which is to go North (the payoffs are equal for one strategy but strictly better for the other). Since only one of them has a dominant strategy, there is no dominant strategy equilibrium. We must then proceed by eliminating dominated strategies. As we’ve already mentioned, for the Japanese strategy ‘go North’ weakly dominates strategy ‘go South’. Therefore, we eliminate the strategy ‘go South’ for the Japanese, who will go North. Now that we only consider the Japanese going North, Kenney’s strategy ‘go North’ is strictly dominant over strategy ‘go South’, which will be eliminated. Therefore, North-North is the weak-dominance equilibrium.

However, as it was mentioned before, it can be easily seen that the game has lost its strategic nature.

Nash equilibria are defined as the combination of strategies in a game in such a way, that there is no incentive for players to deviate from their choice. This is the best option a player can make, taking into account the other players’ decision and where a change in a player’s decision will only lead to a worse result if the other players stick to their strategy. One of the best known Nash equilibriums can be found in the prisoner’s dilemma. This concept belongs to game theory, specifically to non-cooperative games, and was named after John Nash who developed it.

There are a few consistency requirements that must be taken into account when dealing with Nash equilibria. One of them is known as common knowledge, which extends the necessity of complete information. Therefore, expectations about other player’s strategies must be rational.

A Nash equilibrium is therefore a combination of beliefs about probabilities over strategies and the choices of the other player. It is quite easy to understand this using an example, in this case the prisoner’s dilemma as depicted in the adjacent game matrix.

Prisoner 1 (P1) has to build a belief about what choice P2 is going to make, in order to choose the best strategy. If P2 confesses (P2_C), he will get either -8 or 0, and if he lies (P2_L) he will get either -10 or -1. It can be easily seen that P2 will choose to confess, since he will be better off. Therefore, P1 must choose the best strategy given that P2 will choose to confess: P1 can either confess (P1_C, which pays -8) or lie (P1_L, which pays -10). The rational thing to do for P1 is to confess. Proceeding inversely, we analyse the beliefs of P2 about P1’s strategies, which gets us to the same point: the rational thing to do for P2 is to confess. Therefore, [P1_C, P2_C] is the Nash equilibrium in this game (underlined in red).

Nash equilibria can be used to predict the outcome of finite games, whenever such equilibrium exists.  Furthermore, the best equilibrium outcome can be found by using the method of elimination of dominated strategies, which will help us find the best Nash equilibrium by excluding ‘unreasonable’ Nash equilibria.

On the downside, we find the issue that arises when dealing with a Nash equilibrium that is neither social nor ethical, and where efficiency may be subjective, which is the case in the prisoner’s dilemma, where the Nash equilibrium does not meet the criteria for being Pareto optimal (underlined in green). Also, the possibility of multiple equilibria causes the outcome of the game to become less predictable.

The prisoner’s dilemma is probably the most widely used game in game theory. Its use has transcended Economics, being used in fields such as business management, psychology or biology, to name a few. Nicknamed in 1950 by Albert W. Tucker, who developed it from earlier works, it describes a situation where two prisoners, suspected of burglary, are taken into custody. However, policemen do not have enough evidence to convict them of that crime, only to convict them on the charge of possession of stolen goods.

If none of them confesses (they cooperate with each other), they will both be charged the lesser sentence, a year of prison each. The police will question them on separate interrogation rooms, which means that the two prisoners cannot communicate (hence imperfect information). The police will try to convince each prisoner to confess the crime by offering them a “get out of jail free card”, while the other prisoner will be sentenced to a ten years term. If both prisoners confess (and therefore they defect), each prisoner will be sentenced to eight years. Both prisoners are offered the same deal and know the consequences of each action (complete information) and are completely aware that the other prisoner has been offered the exact same deal (therefore, it’s common knowledge).

Description:

Since prisoners cannot communicate and will (supposedly) make their decision at the same time, this is considered to be a simultaneous game, and can be analysed using the strategic form, as in the adjacent game matrix. As described before, if both prisoners confess the crime they will be charged an eight years sentence each. If neither confesses, they will be charged one year each. If only one confesses, that prisoner will go free, while the other will be charged a ten years sentence. These can be seen as the respective payoffs for each set of strategies.

Eliminating all dominated strategies, in order to get the dominant strategy, can solve this game. This is, each prisoner will analyse their best strategy given the other prisoner’s possible strategies. Prisoner 1 (P1) has to build a belief about what choice P2 is going to make, in order to choose the best strategy. If P2 confesses (P2_C), he will get either -8 or 0, and if he lies (P2_L) he will get either -10 or -1. It can be easily seen that P2 will choose to confess, since he will be better off. Therefore, P1 must choose the best strategy given that P2 will choose to confess: P1 can either confess (P1_C, which pays -8) or lie (P1_L, which pays -10). The rational thing to do for P1 is to confess. Proceeding inversely, we analyse the beliefs of P2 about P1’s strategies, which gets us to the same point: the rational thing to do for P2 is to confess. Therefore, “to confess” is the dominant strategy. P1_C, P2_C is the Nash equilibrium in this game (underlined in red), since it is the set of strategies that maximise each prisoner’s utility given the other prisoner’s strategy.

Nash equilibriums can be used to predict the outcome of finite games, whenever such equilibrium exists. On the downside, we find the issue that arises when dealing with a Nash equilibrium that is neither social nor ethical, and where efficiency may be subjective, which is the case in the prisoner’s dilemma, where the Nash equilibrium does not meet the criteria for being Pareto optimal (underlined in green).

Generalisation of the game:

The prisoner’s dilemma is not always presented as we have seen in this case. Payoffs for each set of strategies will vary, depending on each person. However, there are a few rules that can be used to build a “proper” prisoner’s dilemma game.

In the adjacent game matrix, we’ve renamed each player’s payoffs, in order to determine the conditions needed to design a prisoner’s dilemma game. In a traditional prisoner’s dilemma, we have: A > B > C > D (in absolute terms). In our previous example, this condition is met (A=10, B=8, C=1 and D=0). In every case, A>B and C>D imply that confess-confess is a Nash equilibrium.

It must be noted that the asymmetry of the game is not the important part of the prisoner’s dilemma. The interesting thing about this game is the fact that its Nash equilibrium is not socially optimum.

Repeated prisoner’s dilemma games:

In order to see what equilibrium will be reached in a repeated game of the prisoner’s dilemma kind, 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.

Simultaneous games are those where decisions are simultaneous: both we and the other ‘player’ choose at the same time. The simplest example of this is probably ‘rock, paper, scissors’. Complete information means that we know what we stand to win or lose: we know that rock beats scissors and that this will give us some form of utility (we might get to pick the plan for that evening or simply feel a warm glow of superiority). We also know that our opponent has this same information, this is, the rules of the game and each player’s payoffs are common knowledge. Simultaneous games are usually described using the strategic (or normal) form.

What this all means is that we only have one chance to get it right, but that we can play smart by knowing what our opponent will do and acting accordingly. An equilibrium is reached when both players will rationally make a decision that they have no reason to change: whatever else they do, they will only be worse off. Our situation can only be improved if our opponent chooses to do something else. These equilibria are known as Nash equilibria after John Nash, an economist from the early/mid 20^th century. The most famous example of this is the prisoner’s dilemma.

In this game, two prisoners are locked up in separate cells where they have no chance of communicating (imperfect information). They are both given the same (complete) information, which is summarised in the following game matrix: 

In this example, the outcomes represented are years of freedom lost. If both prisoners confess, they will both receive an 8 year sentence. However, if neither confess, charges will have to be mitigated and both will receive low sentences. If one confesses and the other doesn’t, only one will receive punishment. Obviously, the best social outcome is for neither to confess, which would be the Pareto optimal situation. However, we can see that this is not a stable equilibrium. Both prisoners have an incentive to confess:  if we are prisoner 2 (the vertical player) and we assume that prisoner 1 is going to confess, we have two possible outcomes: -8 if we confess and -10 if we lie. If we assume that prisoner1 is going to lie, we again have two outcomes: 0 if we confess or -1 if we lie. This means that, independently of the choice our opponent makes, we always have an incentive to tell the truth. Because we know this, we will choose to confess, and, because our opponent is in the same situation, chances are they will do the same. Once at this equilibrium, neither opponent has an incentive to change. This means we have one clear Nash equilibrium: where both players confess.

The process we have just followed in order to reach this conclusion is based on finding a dominant strategy (confess) which always overpowers the alternative under any choice the other player makes (whether our opponent confesses or lies). This allows us to rule out the overpowered strategy. In static games, we can have countless options, but this rational strategy will allow us to see our options more clearly and eliminate undesirable options in a systematic way.

What this process and what Nash equilibria do not allow us necessarily to do are to reach social optimums. The social optimum in this case would be for both players to lie and have charges mitigated: they would both be out within a year. But game theory tells us that social inefficiencies are common when dominant strategies are not for the common good.