Details

Although having been the subject of rather old results, starting from work of Blaise Pascal on the Question of the parts which gave a first intuition of the Probabilité S and mathematical Espérance, and of astonishing sound bet. The game theory became an important branch of mathematics only as from the years 1940, and more especially after the publication in 1944 of the Game theory and economic behavior ( Theory off Ranges and Economic Behavior ) by John von Neumann and Oskar Morgenstern. This work founder detailed the method seen higher of resolution of the plays with null sum.

The theory met at the time of its presentation a sharp opposition on behalf of the staffs: if those randomly accepted readily the use of pullings in the plays of Kriegspiel of the military academies, the idea to give to the fate, in the name of the mixed strategies the fact of escorting or not really such or such convoy hardly filled with enthusiasm those which, resulting from the ground and knowing what were human losses, considered the process at the very least riding.

Towards 1950, John Nash was the first to present a definition of an optimal strategy for a play to several players, said balance of Nash. This brilliant late result was refined by Reinhard Selten; that their was worth the " Nobel Prize of économie" in 1994 for their work on the game theory, with John Harsanyi which had worked on the plays in incomplete information.

Association between play and number by Conway was established in the years 1970.

Broad outlines

The game theory studies the behaviors - envisaged, real, or as justified a posteriori - individuals vis-a-vis situations of antagonism, and seeks to highlight optimal strategies . Very different situations apparently can sometimes be represented with comparable structures of incentive, and constituting as many examples of the same play.

The nonco-operative game theory applies to situations where players play knowingly whereas they have goals at least partially antagonistic (it thus does not apply to the situations of full Coopération, but with the Compétition or his alternative more frequent than one names the Coopétition ). It does not relate to the situations of play against a natural deprived of goals, not drawing up plans, situations where there would be thus makes of it that only one player.

Types of plays

The game theory classifies the sets of it categories according to their approaches of resolution. The most ordinary categories are:

Co-operative plays and competitions

Theory of the negotiation

The modern theory of the negotiation is articulated on the fact that a Négociation constitutes a play with not-null sum. The art of the negotiation thus consists in less making yield the interlocutor on the principal line of opposition (a price, for example) that to find arrangements external with this line which will bring much to the one without being too expensive the other (strategies known as Gain-gaining or ).

For a long time, all that was used in the negotiations:

• “I cannot accept this cost FOB, but then to consider it in CIF. ”
• “If I take two of them to you, do you grant 5% of handing-over to me?”
• “I offer such an amount of of it to you, but it is necessary to decide to you immediately.”

even between private individuals:

• “I want you to leave it well at this price, but you offers the coffee”!

“Coopétition”

Strategy games with null and nonnull sum

• the plays with null sum are all the plays where the sum " algébrique" profits of the players is constant . What gains one is necessarily lost by another, the stake is the distribution of the fixed total, which one can suppose distributed in advance, which brings back if the profits are really null (from where the denomination). The failures or the Poker are plays with null sum because the profits of the one are very exactly the losses of the other.

• the situations of businesses, the political life or the Dilemme of the prisoner are Jeux with not-null sum because certain exits are overall more advantageous for all, or more detrimental for all. One started historically by studying the plays with null sum, simpler. Beyond matter-energy with the law of the conservation of the null algebraic sum, the play with not-null sum is conceivable, in which the profit of the one can benefit the other. Such is the case with information, the communication and the training where information is one of the three basic components with the matter and energy. The illustrative example simplest is the genetic information of DNA transcribed on ARN to be " lue" , " traduite" and to organize the biological matter and energy. In social sciences, one quotes sometimes the ideology of industrial Harmonie of the modern Japan (tripartite coalition capital-work-government) like example of play to nonnull nap. In the international business, the example of this play with nonnull nap is the co-operative competition of the Asian Tigres and Asian Dragons where the profit of the one benefits the other, in the tread of the Japanese Miracle of the Années 1950 - 1960 which opened the doors with the Korea, with HongKong, with Singapore, Taiwan and the Vietnam, in a technical-commercial coévolution.

In ecology, the Coévolution is another example, in nature, of the nonnull sum where the change of the one facilitates and makes the promotion of the change of the other.

One could believe that it would be enough to bring back a play to not-null sum to a play to null sum to add a player simpleton to it, the “table”, kind of Non-player character which would compensate for the net losses of the players. It is not the case: a player is supposed to defend rationally his interests as far as his possibilities; this formal addition, introducing a dissymmetry between “truths” players and the “table”, complicates the analysis and this one loses there more than it does not gain there.

Synchronous or asynchronous play

Repeated plays

For example, it can be useful to take the risk punctually to lose “to see”, test the other players, and to set up communication strategies by the played blows (in the absence of another means of communication).

It also develops phenomena of reputation which will influence the strategic choices of the other players. In the Dilemma of the prisoner, the fact of knowing that one will play several times with hard which never acknowledge but cruelly avenge, or with a coward who always acknowledges, changes the optimal strategy radically.

Lastly, curiously, the fact that the full number of parts is known in advance or not can have effect important on the result, the ignorance of the number of blows bringing closer to the play with an infinite number of blow, whereas its knowledge brings closer contrary to the play with only one blow (and this, as large as is the number of blows!)

Complete information, perfect information

• its possibilities of action
• possibilities of action of the other players
• profits resulting from these actions
• motivations of the other players

In addition one in the case of speaks about play with perfect information play with sequential mechanism, where each player is informed in detail of all the actions carried out before his choice.

The failures are with complete information and perfect. Because of uncertainty on the profits (charts of the adversary hidden), the poker is with incomplete information. The phase of bidding checks the properties of perfect information, but by comparing the pulling of the charts to the action of a fictitious player (often called Nature), the game theory in general excludes the poker from the plays with perfect information.

The real situations are seldom in complete information, and this case is used often only for the trustful approximations.

The plays in information incomplete are strategic situations where one of the conditions is not checked. It can be by the intervention of the chance during the play (frequent case in the board games), or because one of the motivations of an actor is hidden (important field for the application of the game theory to the economy).

The plays in at the same time imperfect and incomplete information are most complex by far. In these plays certain players can have clean information on the way in which the chance will intervene in the exit of the play (a better knowledge of the probabilities of occurrence of such or such event which will affect the course of the play, for example). The war games ( war ranges ) concern typically this category, the risk on the success of an engagement between body of troops depending on information not shared by the adversaries on the power struggles between these troops.

To be complete, it is also advisable to distinguish the plays with perfect memory and imperfect memory . Plays with memory " parfaite" are situations where each player can constantly remember succession of blows which were played previously, to the requirement by progressively noting the played blows. Plays with memory " imparfaite" suppose a kind of amnesia on behalf of the players. The war games are examples of plays to imperfect memory if the commands of operational zones do not manage to communicate between them or with the Staff and thus do not have trace of the movements already carried out by the friendly troops when they must decide their own movements. A typical play is the 21 or blackjack : the convention according to which the continuation of packages of charts is not beaten between two plays can give a light favors with the player since this one takes into account this partial information .

Given plays

Representations of the plays

Extensive form

An extensive form is a decision tree describing the possible actions of the players to each stage of the play, the sequence of turns of play of the players as well as the information which they have with each stage to make their decision. This information is represented in form whole of information . The whole of information forms a partition of the nodes of the tree, each unit corresponding to the whole of the nondistinguishable nodes by the player with a stage of the play. If these units are singletons, i.e. they contain one node of the tree of the play, the play is in perfect information: each player can know constantly where it is located in the tree of the play. In the contrary case, the play is with imperfect information. Incomplete information is represented in the shape of a nonstrategic player: the " Nature" , player who makes by chance certain decisions with such or such stage of the play, directing the continuation of the play towards a certain under-tree of the tree of the play.

Form normal

Definition

principal Article: Play in normal form

A play in normal form is the data of the whole of the players, of the whole of the strategies for each player and the payments associated with any possible combination with strategies.

Tabular representation

If the play comprises only two players and a reasonably restricted number of possible strategies, one can represent the play in the shape of a table named Matrice of the profits.

It is about a table with double-entry which enumerates on each side the possible strategies of the respective players. In the box with crossed of two strategies, one notes the couple of profits of the two players. It is what one names (by convention) the Matrice of the payments.

If the play is with null sum and two players, then one can note only the profits of the first player: those of the second are directly opposite. The table of profits is brought back then to a matrix.

One can, with a reduced number of strategies, to try to represent with a matrix a play with three or four players, but that poses often more problems of interpretation and reading that does not bring answers.

Resolution of a play with null sum

The two players decide simultaneously on their strategy.

Intuitive reasoning

In the same way, the player (2), concerning the opposite of the values of the table, which would reflect in the same way would see as Maximum-Min (A) because of the maximum loss eliminates from 30, but does not allow to slice between (B) and (C), where the maximum loss is of 20. And that Maximum-max classifies the three options by order ascending: With (better possible result: -10) B (+10), C (+20). That would push it to choose (C).

The result would be then ac: the player (2) loses 20 with the profit of (1).

But the player (2) can also try to anticipate the choice of (1). He sees thus that if (1) the maximin plays, itself may find it beneficial to choose (B), which enables him to gain 10.

And so in does its turn player 1 anticipate this deviation and prefers to make (b) for then touching 20? Then (2) should again choose (C): us returned here is with the starting point!

Concept of strategy and mixed balance

No answer is essential. How to leave itself there?

A first possible answer is to play randomly, with an equal probability for all the possible blows, without being concerned with profits. That does not appear optimum, there is to certainly better do.

One second strategy is to try to allot a priori a probability to the actions of the adversary, and to choose the best adapted answer. Thus, if (2) allots a probability 50/50 to the options of (1), it must also play 50/50 (B) and (C). But the adversary is not a die which behaves randomly: he also will anticipate. If it is (1) which reflects, it sees well that it is absurd to suppose that (2) will play (A) in a third of the cases. There still there is to certainly better do.

Introduction of probabilities

Remain to determine which distribution of probability will give the best result: it is ideally that which will maximize the profit hoped for independently of the strategy of the adversary. That amounts bringing back to a problem linear Programming, where the unknown factors are probabilities pi to give to each Oi option, with a single solution for each player in all the plays with two players and null sum.

In the case above, player 1 will hesitate between (A) and (b) by choosing (A) in 4 cases out of 7 is 57  % of the cases. The player (2) will never choose (A), but will oscillate between (B) and (C) by choosing (B) in 4 cases out of 7, that is to say 57  % of the cases. Player 1 will be able to hope for an average profit with each part of 20/7, that is to say 2,85.

These calculations result from the resolution of the linear system by introducing the Lagrangien.

Point-saddle

It is remarkable that this strategic choice remains the best even if the adversary is informed of it.

One is thus brought to introduce the interesting concept, in the mixed strategies, of point-saddle : it is about the optimal choice of probability for the two players: that which deviates some penalizes at the same time (even if this strategy is unfavourable for him, because the others will be it still more). The topic had been foreseen by Auguste Detoeuf: If you have only one risk out of thousand to be convinced of lie, do not lie more once on thousand, because this time will only cancel with it all the others where you said the truth . Detoeuf, responsible industrialist, knowingly avoid specifying that there will be sometimes even favors to lie indeed once on thousand rather than to always say the truth.

The mixed strategies are empirically well-known diplomatic and players of Poker, which know the potential benefit obtained by hiding their plans, even when there is of them one which seems obvious. This idea will strike Philip K. Dick which will devote its novel solar Loterie to him .

Applications

The game theory applies to certain situations of the field of the economy, those where candidates (Oligopole S) exist a reduced number. She seeks the rational strategies in situations where the profits of an actor depend not only on his behavior and the conditions of market, but also of that of the other speakers, which can pursue different or contradictory goals. One finds also applications to him in Political sciences or military Stratégie.

The results can be applied to entertainments (like the television game “ Friend gold Foe ” on a cabled chain specialized with the the United States, Game Show Network ) or to more poignant considerations:

• the Crisis of the missiles of Cuba;
• policies of constitution of convoys of boats in time of war;
• the way of managing a political blow of surprise (Nasser with Suez, de Gaulle with the Quebec, Ieltsine at the time of the putsch, electoral advertisements…) or marketing;
• the fight against the Terrorism.

Professor Thomas Schelling, " Nobel Prize of économie" 2005, specialized in the explanation of the various strategies used ( to use ) in the international conflicts, the such cold war and the nuclear war ( dissuasion. )

Albert W. Tucker for example diffused many interpretations of the Dilemme of the prisoner in the everyday life. biologists used the game theory to include/understand and to envisage the results of the evolution, in particular the concept of balance évolutivement stable introduces by John Maynard Smith into his tests the game theory and the evolution of the fight ( Game Theory and the Evolution off Fighting ). See also its book Evolution and the Theory off Ranges .

It is to be noticed that in Théorie of the evolution, the principal adversary of an individual is not really the whole of its predatory, but the whole of the other individuals of his species and other related species. Like points out it Richard Dawkins, a Brontosaure does not need, to survive, to run more quickly than the tyrannosaure which continues it (what would be impossible for him), but simply more quickly than slowest of its congeneric . Similar phenomena occur in economy. All that joined of the psychological considerations: the conflictuality is related to the resemblance than with the difference.

The Probabilités provide to the game theory a conceptual tool . The Statistiques can feed it in data, and the techniques of optimization to provide him computation results.

Profits and aversion with the risk

In the example in mixed strategy definite higher, the participants in the play were regarded as neutral in the risk. That means that they consider that to have a chance on two to obtain 20 and one chance out of two nothing to have is equivalent to obtain 10.

However, the majority of the people are downpours to the Risque, and prefer the surest exits -- and would accept an additional risk only against one hope of more important profit.

An example of this Aversion to the risk can be noticed during television games. If, for example, one proposes to the candidates that is to say a chance on three to have 50  000  € is 10  000  € undoubtedly, much will prefer the guarantee to change their ordinary. The hoped additional income which is required to compensate for the aversion with the risk is called, in Finance, the Allowance for risk. The subscription of policies insurance (where it is not obligatory) is also justified by aversion with the risk.

It is thus rational to build a measurement of the subjective Utilité

• which is a function of the profit and risk,
• which always satisfies the criterion of neutrality to the risk,
• and which thus corresponds to a table of profits in mixed strategy.

More generally, the utility holds account owing to the fact that the large variations are more significant than the small ones (one buys readily a lottery ticket or Lotto, whose very weak price corresponds to a negligible loss, while the profit would be significant), and that the significance of a variation decrease (there is more difference in utility between a profit of 1.000 and one profit of 1.001.000, that between a profit of 1.001.000 and one profit of 2.001.000, even if the difference is of 1 million each time; a chance in a hundred to gain a million is generally preferred with a chance on thousand to gain 10 million, in spite of the equal hope).

Conversely, it can exist a desire to buy risk or fear : that it is of a lottery ticket or a horror film, the excitation corresponding to a value in itself.

In short, the fact of buying a Lotto or lottery ticket, or of playing in a casino, is justified by two components:

• the Secretion of Adrenalin (as when one will see an action film or that a sport at the risk is practiced)
• the qualitative difference between:
• a probable loss which will pass unperceived,
• a not very probable profit certainly, but which will get if it occurs a qualititatif change . This point was defended in front of the Academy of Science by Emile Borel (in reaction against a tendency of its time to regard only the expectation as function of utility) and is in general allowed since then taking into account its best explanation of the behaviors related on the play and the subscription insurance policies.

Sets of figures

John Conway set up a notation for certain plays and defined operations on these plays, in the hope studied the Jeu of go. Starting from surprising associations of ideas, it insulated a subclass with numerical properties, and led to define the very general class of the surreal numbers. However, in spite of this announced progress, no computer program manages to currently play (2006) go with performances of international player.