GAME THEORY BASICS Definition Reaction Curves Sequential moves' trees MODELLING CONFLICTS Prisoner's Dilemma: Imminent net social loss Distribution mechanisms Shapley value: Fair distribution of incomes inside a coalition TEORIA DE JUEGOS Definición La Matriz de Premios Curvas de Reaccion Arboles de resultados sucesivos Secuencia de movimientos en el juego Juegos de Suma Cero Juegos de Suma No Cero Notación simple y notación extendida en las matrices de premios Resultado de un juego por analisis de la matriz de pagos El concepto de Equilibrio Nash Juego suma-cero con decisiones inciertas El valor de un juego Valor ex ante o estimado Valor ex post o efectivo El juego socialmente justo Análisis de la dominancia de opciones CONFLICTOS COMPLEJOS Dilema del Prisionero: Competencia fatal caracterizada por el aislamiento de los agentes MECANISMOS DE DISTRIBUCION El Duopolio de Cournot: Distribución de mercados obtenida por competencia no cooperativa El Dupolio de Hotelling: Distribución hostil de un mercado My Personal Approach to the Prisoner's Dilemma - 1 There are two investors: Tom and Dick. Each investor wants to acquire stock from the Acme Company and the Bristol Corporation. Each investor knows the intention of his competitor. If Tom and Dick each one buy Acme shares, the value of Acme stock owned by each investor will be 20 million dollars. If Tom and Dick each one buys Bristol shares, then the value of this stock will be that of 10 million dollars. But there are still two more scenarios: if Tom buys Acme and Dick buys Bristol, then Acme will worth nothing (0) but Bristol will soar to 30 million dollars. Last scenario: If only Dick buys Acme and Tom buys only Bristol, then Acme will be worthless and Bristol will reach 30 million dollars. The Prisoner’s Dilemma model describes such situation and proposes a ‘solution’, which is not actually a solution but a forecast of the final outcome of this conflict. So, what do all conventional approaches say about the outcome of this situation? Well, conventional (probably in the same sense as the term 'conventional wisdom') says is that: Tom will buy Bristol and Dick will also buy Bristol. Does this make sense? It should, for 100% of theorists will state that the corresponding scenario describes the final outcome of the game. Dick buys Acme Dick buys Bristol Tom buys Acme 20, 20 (win-win) 0, 30 Tom buys Bristol 30, 0 10, 10 (the 'trap') So, our two savvy investors will follow their drive for money and will fall in a trap, for instead of earning each one a stock value of 20 million dollars, each one will reach a slim value of only 10 million dollars. How smart is this kind of behavior? Do we agree that real-life investors reason that same way? Wait, just give me a minute, I feel that there is something wrong in this standard/conventional approach to the Prisoner's Dilemma situation. Marginal Utility Theory may state something like this: the first 10 million dollars are to be more appreciated than the second 10 million, and these also will be assessed in more welfare than the third 10 million dollars. This I can use as a tangential support for own viewpoint: that if investors secure 20 million dolars each one and the can gain additional 10 million, the gain of this 10 million plus will be less appreciated than the quality of protecting themselves against a loss of 10 million dollars. So each investor will likely seek the 20-20 scenario and discard any possibility of converting those 20 in 30 which may imply a possibility of losing 10 (converting those 20 in 10). This protection quality is more appealing than the growth quality for when money is less, its value increases. As I say, this is only a tangential consideration, and not the core of my view. So I just removed the emotion of the situation of two prisoners in separate cells, and somehow replaced it with the emotion of two millionaires seeking to expand their wealth. It seems that the Prisoner's Dilemma model is not a universal approach to different kinds of interpersonal conflicts, but that its "hardware" (the matrix and the intepretation mechanisms) requires a special "software", likely to include certain attitudes, certain emotions. The question, again, is: what are Tom and Dick supposed to do now? Follow each one his own drive for gain and decide to buy Bristol, both of them falling in a trap? Or is there any possibility that the two of them, without making any kind of contact, decide, each one, to buy Acme? Well, we'll see. My Personal Approach to the Prisoner's Dilemma - 2 The Prisoner’s Dilemma says that two prisoners put in isolated cells will use a criterion of getting maximum possible welfare in order to decide if they confess the mutual crime or just keeps the mouth shut. Part of the solution of this model, according to my own view, relies on the power of emotion, so we resort to emotion and recreate the problem inside our minds. Pure mathematics are neutral and devoid of emotion, but the mental image of a prisoner seeking his way out is enough to disturb my thinking, and likely also yours and anybody else’s. So let’s not pretend that the solution for the Dilemma is just a super-logical one: it is not. I am using a trick when reviewing the Dilemma: I replace the fear for being in prison with the prospect of a large income to be earned. Delinquents aren’t always the brightest people in the world, also, so let’s substitute them by two rich men. Tom and Dick, millionaires, need each one to decide whether he will purchase one company or the alternative one. The 2x2 table of outputs replicates the problem for the prisoners. The motivation (or the subjacent emotion) is not of fear, not even of greed, but of the drive to win. These two guys are really smart, and so I have faith that they will attain the win-win scenario. By means of which criteria could they do so? Then, I drop the idea that they will reach the win-win box, and simply start analyzing the situation, probably the way a chess player will. I split the one output table in two, from the viewpoint of Tom (the payoffs are the same, and it will be clear that in this case solving the problem for one doesn’t really require solving simultaneously solving the problem for the other). First table for Tom is prizes, and this is built using a net and percentage transformation of the payoffs he can get. Second table for Tom is probability, built upon a net and percentage transformation of the payoffs Dick can get. We have then produced the following two tables: Dick buys Acme Dick buys Bristol Tom buys Acme 33.33% (win-win) 0% Tom buys Bristol 50% 16.67% (the 'trap') Dick buys Acme Dick buys Bristol Tom buys Acme 33.33% (win-win) 50% Tom buys Bristol 0% 16.67% (the 'trap') But this is not enough. The last step is to synthesize the two tables, and this I do by cell by cell multiplication, and also transform the resulting figures (which are fractions) in whole numbers (just for ease). Then we get this payoff-matrix for Tom: Dick buys Acme Dick buys Bristol Tom buys Acme 4 (win-win) 0 Tom buys Bristol 0 1 (the 'trap') And this is the last table, the, according to me, one and only intelligent outcome of the, so called, Prisoner’s Dilemma. The win-win scenario comes by itself. Rather than just labeling it a “Prisoner’s” we will acknowledge the dilemma of the election of prizes. It is said that a big prize unlikely to be won is not as preferred as a smaller prize very likely to be got.