Game Theory - Introduction The decision-making process in situations where outcomes depend upon choices made by one or more players. The word "game" describes any situation involving positive or negative outcomes determined by the players choices and, in some cases, chance.
Game Theory - Evolution 1921 - Emile Borel, a French mathematician, published several papers on the theory of games using poker as an example. 1928 - John Von Neumann published his first paper on game theory in 1928, is made it popular. 1944 – Theory of games and Economic Behavior by John von Neumann and Oskar Morgenstern is published. 1950 – Prisoner‟s Dilemma is introduced, introducing the dominant strategy theory. 1953 – Solution to non-cooperative games was provided with the evolution of the Nash Equilibrium. 1970 – Extensively applied in the field of biology with the development of „evolutionary game theory‟. 2007 – Used in almost all the fields for decision making purposes, including the software to track down the terrorists.
Game Theory - Assumptions Each player is rational, acting in his self-interest; The players choices determine the outcome of the game, but each player has only partial control of the outcome; Each decision maker has perfect knowledge of the game and of his opposition;
Game Theory - Classification Single Player v Multi Player Games Co-operative v Non-Cooperative Games Symmetric v Asymmetric Games Zero-sum v Non-Zero-sum Games Simultaneous v Sequential Games Perfect Information v Imperfect Information
Single Player Game – Gamesagainst Nature The outcome and the player‟s payoff depends on both his chosen strategy and the “choice” made by a totally disinterested nature. A Game Against Nature part of what is generally called decision theory (rather than game theory) because there is only one player who makes a rational choice and is interested in the outcome.
Multi Player Games - Examples Prisinor‟s Dilemma Travellers‟ Dilemma Battle of the Sexes Diners‟ Dilemma Rock, Paper, Scissors!!!
Prisoners‟ Dilemma Both the prisoners are more likely to defect irrespective of what the other prisoner does, even though it gets them a sub-optimal output. If they were allowed to communicate and reach a consensus, then they could have reached the optimal output. Prisoner B stays silent Prisoner B confesses (cooperates) (defects)Prisoner A stays silent Prisoner A: 1 year Each serves 1 month (cooperates) Prisoner B: goes freePrisoner A confesses Prisoner A: goes free Each serves 3 months (defects) Prisoner B: 1 year
Battle of the Sexes A couple had agreed to meet in the evening but had not agreed on the venue and cannot communicate now. They can either go to the opera or the football match. Their pay-off matrix can be Football by – Oper given a Opera 3,2 1,1 Football 0,0 2,3
Diners‟ dilemma It is a n-person‟s prisoners‟ dilemma. A group of individuals go out to dine together. They agree that they will split the cheque equally between them. Each individual must now decide whether to order the cheaper dish or the expensive one. It is presumed that the exensive dish is better than the cheaper ones but the price differential is not justified.
Diners‟ dilemma - Consequences Each individual reasons that the expense which they add to their bill while ordering the more expensive item is very low. Hence, they justify the cost to experience the improved dining experience. However, each individual reasons similarly, and thus they all end up paying for a more expensive dish. By assumption, this is worse than each of them ordering and paying for the cheaper dish.
Rock, Paper, Scissors!!! It is a two-player zero-sum game. No matter what a person decides, the mathematical probability of his winning, drawing, or losing is exactly the same. The dominant strategy to this game seems to exists, which is why the same person end up in the merit roll of the championships held around the world every year. Child 2 rock paper Scissors rock 0,0 -1,1 1,-1Child 1 paper 1,-1 0,0 -1,1 scissors -1,1 1,-1 0,0
Travellers‟ Dilemma Case designed by Dr. Kaushik Basu in 1994. Each traveller can value there belongings for anything between $2 and $100. They will be reimbursed the lower value of the two claims. The lower claimant will be rewarded with additional $2 while the higher claimant will be charged $2.
Travellers‟ Dilemma - Paradox The rational strategy for the travellers would be to claim the lower value, i.e. $2. In reality, people chose $100, which resulted them in being better-off financially. This experiment rewards people for deflecting from the Nash Equilibrium and act non-rationally. This has led people to question the practicality of the game theory. Subsequently, the idea of „super-rationality‟ was developed under this, which stated that under pure strategies $100 is the optimal solutioin for the problem.
Strategies Dominant Strategy – a strategy which dominates irrespective of what the other player does. Maximax Strategy – The player looks to maximize the maximum pay-off that he may stand to gain from the game. Minimax Strategy – The player looks to maximize the minimum payoff that he receives. Collusion – When both players decide to co-operate to maximise their total output. Tit for tat – A player reacts to the opponents actions by following it, i.e deflection followed by deflection. Backward Induction – The player derives his strategies by working the most likely strategy of his opponent and then working backwards. Markov Strategy – A strategy through which a player decides his actions based only on his present state, ignoring the past states.
Nash Equilibrium It is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. It does not necessarily mean the best pay-off for all the players involved although it might be achieved as is the case with cartels.