iterative dominance in game theory

8/10/2019 Iterative Dominance in Game Theory

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Submitted By:Himasagar Reddy(11BCE0277),

Nandan Babu(11BCE0085),Rohit Enduri(11BME0426)

Under the guidance of:

Prof. Sundaramali G


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Abstract: We can often make sharper predictions about the possible outcomes of a

game if we are willing to make stronger assumptions. Up until now we have

assumed that the players are rational but we haven't even assumed that

each knows that the others are rational. Beyond that we could further

assume that each player knows that the other players know that the othersare all rational. We could continue adding additional layers of such

assumptions ad nauseam. We summarize the entire infinite hierarchy of

such assumptions by saying that the rationality of the players is common

knowledge.

Rationality constrains players to choose best responses to their beliefs but

does not restrict those beliefs. Common knowledge of rationality imposes aconsistency requirement upon players' beliefs about others' actions.


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By assuming that the players' rationality is common knowledge, we canjustify an iterative process of outcome rejection--the iterated elimination of

strictly dominated strategies--which can often sharpen our predictions.

Outcomes which do not survive this process of elimination cannot plausibly

be played when the rationality of the players is common knowledge.

A similar , and weakly stronger, process--the iterated elimination ofstrategies which are never best responses--leads to the solution concept of

rationalizability. The surviving outcomes of this process constitute the set of

rationalizable outcomes. Each such outcome is a plausible resultandthese

are the only plausible resultswhenthe players' rationality is common

knowledge. In two-player games the set of rationalizable outcomes is

exactly the set of outcomes which survive the iterated elimination of strictlydominated strategies. In three-or-more-player games, the set of

rationalizable outcomes can be strictly smaller than the set of outcomes

which survives the iterated elimination of strictly dominated strategies.


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In a rationalizable outcome players' beliefs about the same question can

differ--and hence some are incorrect; and a player can find--after the others'

choices are revealed--that she would have preferred to have made a

different choice.

So the problem can be brought down to a 2x2 matrix which can be solved

by few formulae.


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Objective: The result that obtained from combination of players strategies . Every

combination of strategies is an outcome of a game. A primary purpose of

game theory is to determine which outcomes are stable.

Usefulness of Game Theory:

Computer science: Game theory has come to play an increasingly important role in logic and

in computer science. Several logical theories have a basis in game

semantics. In addition, computer scientists have used games to

model interactive computations.


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Computer science: Also, game theory provides a theoretical basis to the field of multi-agent

systems.Separately, game theory has played a role in online algorithms. In

particular, the k-server problem, which has in the past been referred to

as games with moving costsand request-answer games(Ben David,

Borodin & Karp et al. 1994). Yao's principle is a game-theoretic techniquefor proving lower bounds on the computational complexity of randomized

algorithms, and especially of online algorithms.The emergence of the

internet has motivated the development of algorithms for finding equilibria in

games, markets, computational auctions, peer-to-peer systems, and

security and information markets. Algorithmic game theory]and within

it algorithmic mechanism design]combine computational algorithmdesign and analysis of complex systems with economic theory.


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Applications of Game theory:

The situations to which game theory has actually been applied reflect

its selective usefulness for problems and solutions of an individualistic and

competitive nature, building in the values of the status quo. The two

principal areas of application have been war and economics. For the

military it has been applied to tactical decision-making (in particular via the

theory of differential games) and in studying global nuclear strategies such

as deterrence. In economics, game theory has been used in studying

competition for markets, advertising, planning under uncertainty, and so

forth. These primary areas of application - war and economics - are where

one would expect game theory to be applied, given the values reflected inits concepts.


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Game theory has also been applied to many other fields, such as law,ethics, sociology, biology, and of course parlour games. In all these

applications, a close study of the formulation of the problem in the game

theory perspective shows a strong inclination to work from existing values,

consider only currently contending parties and options, and in other ways to

exclude significant redefinitions of the problems at hand. Presently I will

give examples of this inclination, but first it is worth mentioning the principal

uses of game theory.


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Although game theory has been applied to many situations, it has notbeen particularly fruitful - at least in terms of its original promise. I seeat least three ways in which game theory has proved 'useful'. First, ithas to led to practical advice on tactical decision-making in certain well-defined situations, especially in military areas involving missile trackingand similar tasks (where the theory of differential games has led to

results equivalent to control theory).Second, it has provided anoccupation and amusement for thousands of government bureaucrats,mathematicians, psychologists, and others who have found plenty offunds to study game theory, develop its mathematical ramifications, andplay around with bargaining and simulation games. Third, it hasprovided a perspective for looking at military and political choices that

builds in many values of the status quo, that can be adapted to givenearly any results desired, and which has the appearance ofmathematical sophistication. Game theory formulations therefore serveadmirably as ex post facto justifications for any decisions or policiesthat may be adopted by military or political lites.


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The values built into game theory concepts thus seem to be closelyreflected in its areas of primary application (war and economics) and in

what it has actually been used for (tactical decision-making, employment of

people studying game theory, legitimizing military and political decisions).

Until now the impression may have been given that game theory is primarilyused to represent (for academics) the way in which decisions are made. But

in many cases game theory is used as a tool by certain people who are

actually in these situations. In these applications, the ostensible reason for

applying game theory is to obtain insights concerning what policies should

be adopted by particular actors. One important actual result of such

applications, though, is an implicit justification and reinforcement of theassumptions which are built into the game theory formulation itself.


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That is, by specifying a limited range of potential action, game theory

formulations encourage a perception that these actions are the only

feasible or rational ones.

The following case studies illustrate the type of narrow perspective that

inevitably seems to arise when game theory is applied to a problem

situation. The areas I briefly comment on are international relations,

ethics, and crime. Applications in other areas are similarly limited.


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Program code:#include

#include

using namespace std;

//xxovoid print(int a[]);

//int n,m;

int main()

{

int n,m;

coutm;

int a[n+1][m+1];

cout


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for(i=0;i


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if(count==m){

for(j=0;j


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if(a[i][j]


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for(i=0;i


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Enter value for player A 1 and for B player 2-1

Enter value for player A 2 and for B player 1-1

Enter value for player A 2 and for B player 20

SUM R 4

SUM C 4

Row 1 0.75

Row 2 0.25

Col 1 0.75Col 2 0.25


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Player A1 with B1 is -0.25




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19/19

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iterative dominance in game theory

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