**By Morgan Donnette.**

Game theory has become, during the last century one of the best ways in the decision-making process. Not only in economics but also in the everyday decisions and, furthermore, for the decision related to the entrepreneurship decision making related the place on the market of a company. Hence, we are going to see today an introduction to Game Theory and a reflexion on the use of Game Theory in the context of a free market.

I – Introduction…………………………………………………………………………………………………………….

II – The exclusion of notion of infinity in matrix calculation…………………………………………….

III – The application of an extensive form of Game theory in a limited free market………….

IV – The invisible hand maker of Nash equilibria?………………………………………………………….

V – Conclusion…………………………………………………………………………………………………………….

VI – Appendix:……………………………………………………………………………………………………………..

Appendix – A : Ramanujan summation mathematical demonstration……………………………………

Appendix – B: Prisoner’s Dilemma two players’ explanation………………………………………………

VII – References:………………………………………………………………………………………………………….

# I – Introduction

Adam Smith, the father of Economics said in the book the Wealth of Nations that: “It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our necessities but of their advantages” (Smith, 1776, Book I, p. 26). As Adam Smith said, in a market, everyone tries to make as much profit as possible, not for the benefits of the community but for the benefits it might bring for himself. “Selfishness” in economics might be beneficial for society. At the end of the book, Adam Smith goes even further: “Every individual… neither intends to promote the public interest, nor knows how much he is promoting it… he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention.” (Smith, 1776, Book IV, p. 456) At this moment Adam Smith introduces a concept “the invisible hand”. To explain this concept in a few words, in a free market, the market will always by itself find an equilibrium; if every individual tries to do what is the best for himself, so, the market will by itself make a balance between, on one side, demand and, on the other, supply.

During more that 200 years the concept of Adam Smith was considered as the main rule of economics: “In competition, individual ambition serves the common good”. Hence, if anyone in a market does what is best for himself so they do the best for the common good. However, in the 50’s, a young mathematician John F. Nash, Jr. revolutionised this thinking pattern. His idea: The Best for the Group comes when everyone in the group does what is best for himself AND the group. In a few years, he published different papers that would revolutionise the economics pattern. (Nash, 1949-1953). John Nash discovered what we call today today the “Nash equilibrium” in Game theory. To understand this new concept, it seems necessary to have a first look at what is Game theory. The first time the term “Theory of Games” was use was in 1944, in an article written by Von Neumann and Morgenstern, (1944). It was an article taking up the ideas of another article by John von Neumann, “Zur Theorie der Gesellschaftsspiele” (1928) literally “the theory of social games”. But if Game theory is known today it is because of John Nash’s discovery of the Nash equilibrium. However, we firstly have to give a definition of what is Game theory. According to Roger Myerson, Game theory is « the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory provides general mathematical techniques for analysing situations in which two or more individuals make decisions that will influence another’s welfare. » (Myerson, 1997, p1). John Nash’s main discovery, the Nash equilibrium, is “a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent’s choice.” (Analoui & Karami, 2003 p143). According to these definitions, Game theory, in economics seems not to be applicable in a free market economy, it seems that the game theory applies only in situations where there is strategic interdependence between the players. Thus, to calculate a Nash equilibrium, a limited number of variables is necessary. And in game theory every player has to take into consideration all the different possibilities according to the choice of the other player, and because of this interdependence, each player will have a payoff that will depend not only on his own action but also on the action of the other players. Therefore, it seems unrealistic to apply this theory to a free market, because in a free market, “the government does not intervene, and leaves all decisions to individuals and private firms to work out for themselves.” (Gillespie, 2014, p9) In a free market, one element is important: Competition. The main problem to have a competition, you need a high supply, but to have a perfect competition, you need all the agents to be price takers. In other words, no agents can change the price at all, and to have a real equality, the only way to have only price takers would be to have an infinite demand and an infinite supply. So actually we have two different approaches to find an equilibrium depending if we are in a free market so the invisible hand will apply or if we are in an oligopolistic market so the Game theory applies. So, can we go overcome the apparent impossible application of the game theory in a free market: Can we make an end of the monopoly of the invisible hand in a free market?

We will try in this essay to prove the possibility – or the impossibility – of calculating a Nash equilibrium or even some Nash equilibria in a free market. For this reason, we will see firstly the exclusion of the notion of infinity in matrix calculation and therefore in game theory, secondly the application of an extensive form of game theory in a limited free market and, finally, see if the invisible hand could create a Nash equilibrium or even equilibria.

# II – The exclusion of notion of infinity in matrix calculation

As we said previously, one of the major problem to applied the game theory in a free market is the necessity to have a perfect competition with an infinite number of demand and supply. Even, if in reality there will never be an infinity of demand and supply, let us try to see if it is possible to found an equilibrium when the numbers are infinite.

Here, the question is more mathematical than economic, to calculate a Nash equilibrium the basic form of representation is the matrix. It helps to represent the different possibilities and to establish a Nash equilibrium or even some Nash equilibria. There also are some extensive forms of representation of a game. But to facilitate the task we will concentrate here on the matrix; can we calculate a matrix with an infinity of numbers or an infinity of dimensions? The answer could seem obvious, mathematically the creation or the resolution of a matrix with an infinity of dimensions is possible under some conditions (Gaudin, 1988). However, we are not looking for a strict mathematical application of the resolution of a matrix, we try to solve a resolution of a game, and apply this to an economic field. The problem in economics is that we cannot always apply the mathematical concept *stricto sensu* (strictly speaking). For example, one way to solve a situation, in our case a matrix, with the management of infinite numbers could have been to use the Ramanujan summation to deal with it. However, if it works in physics for example in the *string theory*, and even if we know how to deal with some divergent infinite series, like , we cannot apply this series in economics for one good reason, economics is not a pure mathematical concept but a created concept where some mathematical concepts cannot be applied. For example, in a matrix, if we arrive to a function where the calculus includes one of the Ramanujan summation we could arrive with a sum of positive numbers to a negative equilibrium. If mathematically it is not a problem, in economics the equilibrium could never be a negative price. But maybe the reason why an equilibrium could not be negative is because the notion of infinity in economics is a pure fiction. Because the number of resources is limited, the needs are never infinite, and because the number of people on earth are also limited. A true infinity in economics is an illusion because we will always have to deal with finite numbers, we can imagine a large number of actors, a large number of players maybe even billions, but never an infinity.

# III – The application of an extensive form of Game theory in a limited free market

Now that we have established that the concept of infinity is an illusion in economics, we can suppose and speak about a ‘limited free market’, more realistic. In this limited free market, the number of players can be huge, but never infinite. Now that we agree on a definition for “free market”, we can come back to Game theory. Firstly, in game theory, what is a “game”? “A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence.” (Mas-Colell, Whinston and Green, 1995, p219) In other words, in Game theory, the well-being of the individuals is not only determined by the choice they made, but also by the other individuals’ choices. We are here in “multiperson situations with strategic interdependence” (Mas-Colell, Whinston and Green, 1995, p217).

Consequently, we have to establish which game could fit a free market. And to make this demonstration we firstly have to make the difference between the cooperative game, and the non-cooperative game. To give a definition, “Cooperative game theory focuses on how much players can appropriate given the value each coalition of player can create, while non-cooperative game theory focuses on which moves players should rationally make” (Chatain, 2014). We are in consequence here in a non-cooperative game theory.

We need to analyse first how to calculate a Nash equilibrium in a small market, an oligopolistic market, and then try to apply the same pattern in a bigger market. Thus, let us start with a basic in Game theory; the Prisoner’s Dilemma. (See appendix: B Prisoner’s Dilemma 2D explanation), but let us imagine now that we go further. We do not have two players anymore but three players. Can we still calculate a Nash equilibrium? Are we in a situation where there is now a multiple of Nash equilibria?

Because we are here in a situation of three players we cannot represent the game in two dimensions the matrix, in fact, the matrix for this game is now composed of 8 small cubes that make a bigger cube this cube is our matrix. Nevertheless, we can anyways make a list with all the possibilities and the outcomes. Here the choices are basically binary ether confess (C) or not confess (NC). Let us say that here the rules are:

– If all three players confess, each player will remain 5 years in jail.

– If all three players refuse to confess, each player will remain 1 Year in jail.

– If only one player confesses, he/she walks, and the other two will remain 10 years in jail.

– And, if two players confess, they get 6 years each, and the third player gets 10 years.

So we have:

(C, C, C) = (5, 5, 5)

(C, C, NC) = (6, 6, 10)

(C, NC, C) = (6, 10, 6)

(C, NC, NC) = (0, 10, 10)

(NC, C, C) = (10, 6, 6)

(NC, C, NC) = (10, 0, 10)

(NC, NC, C) = (10, 10, 0)

(NC, NC, NC) = (1, 1, 1)

Here we have a Nash equilibrium ((NC, NC, NC) = (1, 1, 1)), we could have continued through and seen for each player which strategy is dominant and which strategy is dominated; But it is not here the point. The point here is, adding a dimension has changed nothing, we still have a Nash equilibrium. We could have added another player and add on another dimension, it would not have been a problem. The only thing is the more we add dimensions more difficult it is to achieve the probability of a cooperation between the different players to apply the Nash equilibrium. We could continue with four dimensions, five, six… Or ever 1 million dimensions, (but not an infinity because we previously exclude the infinity). So we can basically say that in this game, a Nash equilibrium exists for every game where we have “N” dimension with “N” above two excluding infinity .

The problem with this game to apply it to a free market is, even if we continue to add players until a very big number, each player has to be omniscient and has to have a complete information in the game. We are here in a situation of a complete information but in a bigger market, and to arrive at a potential free market (even in a limited one), we have to find another way to play, a game where the players do not have all the information. This game with incomplete information is called a Bayesian game. To give a definition of what is « Bayesian Game theory » is a game with incomplete information, a game where the players do not have all the information about the other players but, they have beliefs with known probability distribution. Here is the problem, the presence of incomplete information raises the possibility that we may need to consider a player’s beliefs about other players’ preferences, his beliefs about their beliefs about his preferences, and so on, much in the spirit of rationalizability (Mertens and Zamir, 1985). However, as highlighted Mas-Colell, Whinston and Green there is a widely used approach to this problem, originated by Harsanyi (1967-68), that makes this unnecessary. In this approach, one imagines that each player’s preferences are determined by the realisation of a random variable. Although the random variable’s actual realisation is observed only by the player, its *ex-ante* probability distribution is assumed to be common knowledge among all the players. (Mas-Colell, Whinston and Green, 1995, p254).

Without going through all the details let us see what could be the consequences of the two precedent points. Firstly, we have established that mathematically a Nash equilibrium exists under some conditions even in very large games, secondly with Bayesian game Theory, we are able to deal with situations where the players do not have all the information so at least theoretically it seems possible to create a game as big as every market in the world notwithstanding the size of the market, and find a Nash equilibrium.

# IV – The invisible hand maker of Nash equilibria?

As we said previously, we can calculate an equilibrium notwithstanding the size, or even the fact that the players do not have all information in the market. The difficulty lies in the fact that to find the equilibria in a free market competition you need a large quantity of information and the compilation of all of it. The compilation, of all this information, seems complicated but possible; the difficulty seems to be material. Even if an equilibrium exists in a big market, if it cannot be found, or if it can only be found by few players, its utility becomes relative. A Nash equilibrium could be useful only if all players in the game can find it, and following it, it needs the cooperation of all the protagonists of the game or at least a majority to apply it.

But anyway now that we have established that an equilibrium or some equilibria can exist in a free market, can we determine them? Let us here make some suppositions. If we imagine a game which would have a huge size, a lot of players… Every player is going to look for the best response to the other players strategies, but the more you add players in the game, the smaller the gap between player strategies is going to be. Because even the different possibilities for the players start to be reduced owing to this large amount of players, for example the more you add players in the prisoners’ dilemma, the lower the probability that everyone will not confess, so even if the Nash equilibrium still exists when you arrive at a large amount of numbers, for example, the same game but with hundreds of players or even a thousand, the probability that everyone keeps the secret and does not confess is of 1 in 1000 but all the other possibilities are that at least one person confesses. In this situation, because of these probabilities, not confessing is becoming a weak strategy.

So at this point, the best strategy rather than confess is to not confess. This is in a bigger game what seems to be the dominant strategy. But when we arrive in a market where the supply is almost infinite, the first Nash equilibrium tends to disappear and at this point the best play is *post hoc ergo propter hoc* (after this, therefore because of this) what seems to be best for himself. So, the invisible hand rather than the Nash equilibrium seems to become the dominant way to find an equilibrium. But, is this solution a new Nash equilibrium? The invisible hand could be a natural way to find one of the Nash equilibria? No, because, even with a million players for example in the prisoner dilemma, even if the probability of having a Nash equilibrium is 1/1,000,000 so the chances to lose by playing the Nash equilibrium are 99.9999% the Nash equilibrium still exists. If the conditions are the same so the Nash equilibrium stays the same independently of the number of players. But let us now imagine that one player calculates this Nash equilibrium and tries to make the other players aware of the existence of the last one. So the strategy can change and the probability could be change at the same time, and so the dominant strategy could be a new Nash equilibrium even in a free market. The key to applying a Nash equilibrium in a free market seems to be the communication to all players of this last one.

# V – Conclusion

*In fine*, we can say that if a Nash equilibrium exists even in a free market the use of this equilibrium seems to be difficult. Firstly because of the difficulty to finding it, even if you do not need all the information, you need to collect a lot of information and, once the information is available, another difficulty is the application and the calculus to find Nash equilibrium. And, even when the Nash Equilibrium is known, it is complicated to applied and coordinated the market to make the application of this equilibrium. That is why the invisible hand is nowadays still in a free market the best way and actually the only way to find an equilibrium in Free market competition. But we cannot exclude that some Nash equilibria could be found in some market and the publication, the sharing of this could be a solution to have even in a context of a free market the application of some Nash equilibria.

**By ****Morgan Donnette****, editor and Co-founder of Counselution**

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# VI – Appendix:

## Appendix – A : Ramanujan summation mathematical demonstration

## Appendix – B: Prisoner’s Dilemma two players’ explanation

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

- If A and B each betray the other, each of them serves 5 years in prison
- If A betrays B but B remains silent, A will be set free and B will serve 10 years in prison (and vice versa)
- If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)

So the Matrix:

Prisoner A | |||

Prisoner B |
Silent | Betray | |

Silent | (-1,-1) | (-10,0) | |

Betray | (0,-10) | (-5,-5) |

Here we can see that if we applied the Adam Smith theory, each player do what is best for himself, so they both confess, and both of them will go 5 years in jail. But with the Nash equilibrium. By watching the other player’s possibility, and we see that each player shall remain the silent. By this reflexion, each player will only go 1 year in jail.

# VII – References:

Analoui, F. and Karami, A., (2003). *Strategic management in small and medium enterprises*. Cengage Learning EMEA.

Chatain, O., (2014). Cooperative and non-cooperative game theory. From the SelectedWorks of Olivier Chatain

Gaudin, M., (1988). Matrices R de dimension infinie. *Journal de Physique*, *49*(11), pp.1857-1865.

Gillespie, A., (2014). *Foundations of economics*. Oxford University Press, USA. Third Edition.

Mas-Colell, A., Whinston, M.D. and Green, J.R., (1995). *Microeconomic theory* (Vol. 1). New York: Oxford university press.

Mertens, J.F. and Zamir, S., (1985). Formulation of Bayesian analysis for games with incomplete information. *International Journal of Game Theory*, *14*(1), pp.1-29.

Myerson, R.B., (2013). *Game theory*. Harvard university press.

Nash, J.F., (1949). Equilibrium points in n-person games. *Proceedings of the national academy of sciences*, *36*(1), pp.48-49. (1950). The bargaining problem. *Econometrica: Journal of the Econometric Society*, pp.155-162. (1951). Non-cooperative games. *Annals of mathematics*, pp.286-295. (1953). Two-person cooperative games. *Econometrica: Journal of the Econometric Society*, pp.128-140.

Smith, A., (1776). The Wealth of Nations (W. Strahan and T. Cadell, London).

Von Neumann, J., (1928) « Zur Theorie der Gesellschaftsspiele, » Math. Ann., 100, 295-320

Von Neumann, J., and Morgenstern, O., (1944) *Theory of Games and Economic Behavior*, Princeton University Press, Princeton, 1944, pp. 153-155.