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Table of contents
Introduction 2. Arithmetic 3. Introduction to algebra 4. Graphs and functions 5. Simultaneous linear equations 6. Quadratic equations 7.
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ENGG: Engineering interdisciplinary. FINS: Finance. GEOL: Geology. GEOS: Geoscience. GSBE: Architecture. Hence, this is a 4 year B. Theory of Externalities:. In , the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. RAND pursued the studies because of possible applications to global nuclear strategy. Nash proved that every finite n-player, non-zero-sum not just 2-player zero-sum non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.
Game theory experienced a flurry of activity in the s, during which time the concepts of the core , the extensive form game , fictitious play , repeated games , and the Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time. In Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program that cooperates on the first step, then on subsequent steps just does whatever its opponent did on the previous step.
The same winner was also often obtained by natural selection; a fact widely taken to explain cooperation phenomena in evolutionary biology and the social sciences. In , Reinhard Selten introduced his solution concept of subgame perfect equilibria , which further refined the Nash equilibrium later he would introduce trembling hand perfection as well. In the s, game theory was extensively applied in biology , largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy.
In addition, the concepts of correlated equilibrium , trembling hand perfection, and common knowledge  were introduced and analyzed. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening, correlated equilibrium and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
In , Leonid Hurwicz , together with Eric Maskin and Roger Myerson , was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium , and an important graduate text: Game Theory, Analysis of Conflict. In , Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In , the Nobel went to game theorist Jean Tirole.
A game is cooperative if the players are able to form binding commitments externally enforced e. A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing e. Cooperative games are often analysed through the framework of cooperative game theory , which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.
Cooperative game theory provides a high-level approach as it only describes the structure, strategies and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory the converse does not hold provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation.
While it would thus be optimal to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process, or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric.
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The standard representations of chicken , the prisoner's dilemma , and the stag hunt are all symmetric games. Kazuhiro Ohnishi. Book file PDF easily for everyone and every. Some [ who? However, the most common payoffs for each of these games are symmetric. Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player.
It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players. Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero more informally, a player benefits only at the equal expense of others.
Other zero-sum games include matching pennies and most classical board games including Go and chess. Many games studied by game theorists including the famed prisoner's dilemma are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade.
It is possible to transform any game into a possibly asymmetric zero-sum game by adding a dummy player often called "the board" whose losses compensate the players' net winnings. Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions making them effectively simultaneous. Sequential games or dynamic games are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge.
The difference between simultaneous and sequential games is captured in the different representations discussed above.
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