Game Theory - Chapter No 14 - Presentation

7/31/2019 Game Theory - Chapter No 14 - Presentation

1/23

Presented By :Afzal Waseem

Presented To: Dr. Arnold Yuan

*

Ryerson University


2/23

Game theory is a mathematical theorythat deals with the general features ofcompetitive situations like these in aformal, abstract way. It places particular

emphasis on the decision-makingprocesses of the adversaries


3/23

*two-person, zero-sum games

*They are called zero-sum games because oneplayer wins whatever the other one loses, so that

the sum of their net winnings is zer


4/23

*To illustrate the basic characteristics of two-person, zero-sum games,consider the game called odds and evens. This game consists simplyof each player simultaneously showing either one finger or two fingers.If the number of fingers matches, so that the total number for bothplayers is even, then the player taking evens (say, player 1) wins the

bet (say, $1) from the player taking odds (player 2). If the number doesnot match, player 1 pays $1 to player 2. Thus, each player has twostrategies: to show either one finger or two fingers. The resultingpayoff to player 1 in dollars is shown in the payoff table given in Table14.1.In general, a two-person game is characterized by

*1. The strategies of player 1

*2. The strategies of player 2

*3. The payoff table


5/23

*The actual play of the game consists of each player simultaneously choosing a

strategy without knowing the opponents choice.

*In more complicated games involving a series of moves, a strategy is a

predetermined rule e.g Chess

*Player 2 is just the negative of this one, due to the zero-sum nature of the game.

_________________________________________________________

*Entries in the payoff table may be in any units desired, such as dollars, provided

that they accurately represent the utility to player 1.

*The outcome corresponding to an entry of 2 in a payoff table shouldbe worth

twice as much to player 1 as the outcome corresponding to an entry of 1.


6/23

*A primary objective of game theory is the development of rational criteriafor selecting a strategy.

Two Key assumptions are made:

*1. Both players are rational.

*2. Both players choose their strategies solely to promotetheir ownwelfare (no compassion for the opponent).


7/23

*2 Polititians

*Bigtown and Megalopolis

*1 full day in each city or 2 full days in just one of the cities

*Neither politician will learn his opponents campaign schedule until after

he has finalized his own decision

*Best strategy on how to use these 2 days ?


8/23

2 Players

3 Strategies

*Strategy 1 spend 1 day in each city.

*Strategy 2 spend both days in Big town.

*Strategy 3 spend both days in Megalopolis.

*By contrast, the strategies would be more complicated in a different

situation where each politician learns where his opponent will spend thefirst day before he finalizes his own plans for his second day.


9/23

Table represent Utility to Player 1 = Negative Utility to Player 2

Objective : To win votes

Payoff table Entries : Total net votes won from the opponent by Player 1

This payoff table would not be appropriate if additional information

available to politicians*how the populace is planning to vote 2 days before the election, so that

each politician knows exactly how many net votes (positive or negative) heneeds to switch in his favour during the last 2 days of campaigning to winthe election.

*3 Alternative set of games based on Data

*How to solve three different kinds of games?


10/23

Which strategy should each player select ?

The answer can be obtained just by applying the conceptof dominated strategiesto rule out a succession ofinferior strategies until only one choice remains.


11/23

A strategy is dominated by a second strategy if the secondstrategy is always at least as good (and sometimes better)regardless of what the opponent does. A dominatedstrategy can be eliminated immediately from furtherconsideration.

For player 1, strategy 3 is dominated bystrategy 1

Because both players are assumed to be rational, player 2 also can deduce thatplayer 1 has only these two strategies remaining under consideration

strategy 2 for player 1 becomes dominated by strategy 1

Consequently, both players should select their strategy 1.

Player 1 then will receive a payoff of 1 from player 2


12/23

In general, the payoff to player 1 when both players playoptimally is referred to as the value of the game. A gamethat has a value of 0 is said to be a fair game.

Since this particular game has a value of 1, it is not a fairgame.

The concept of a dominated strategy is a very

useful one for reducing the size of the payoff tablethat needs to be considered


13/23

This game does not have dominated strategies, so it is not

obvious what the playersshould do.What line of reasoning does game theory say they shoulduse?


14/23

Both players are rational

Player 1 Strategy 1 winning 6 but loosing as much as -3Player 2 is also rational so he will also avoid his strategy 3So note a good option for player 1Player 1 Strategy 3 large payoff of 5 but more chances of loosing becausePlayer 2 his rational opponent will avoid taking his strategy 1

(This line of reasoning assumes that both players are averse to riskinglarger losses than necessary, in contrast to those individuals who enjoygambling for a large payoff against long odds.)


15/23

Even when the opponent deduces a players strategy, the opponentcannot exploit this information to improve his position. Stalemate

The end product of this line of reasoning is that each player should play insuch a way as to minimize his maximum losses whenever the resultingchoice of strategy cannot be exploited by the opponent to then improve hisposition. This so-called minimax criterionis a standard criterion proposed

by game theory for selecting a strategyNotice the interesting fact that the same entry in this payoff table yields both the

maximin and minimax values.The reason is that this entry is both the minimum in its row and the maximum of its

column. The position of any such entry is called a saddle point.

Because of the saddle point, neither player can take advantage of theopponents strategy to improve his own position

This is a stable solution (also called an equilibrium solution), players1 and 2 should exclusively use their maximin and minimaxstrategies, respectively.


16/23

some games do not possess a saddle point, in which case a morecomplicated analysis is required. How should this game beplayed?


17/23

Player 1 = Strategy 1 gets + 2Player 2 = Strategy 3 looses 2

Player 1 = Strategy 1 looses - 2Player 2 = Strategy 2 gets + 2

Player 1 = Strategy 2 gets +4Player 2 = Strategy 2 looses -4

Player 1 = Strategy 2 looses -3Player 2 = Strategy 3 gain +3

Player 1 = Strategy 1 gets + 2Player 2 = Strategy 3 looses 2

Whole cycle is repeated


18/23

Therefore, even though this game is being played only once,any tentative choice of a strategy leaves that player with amotive to consider changing strategies, either to take advantageof his opponent or to prevent the opponent from taking

advantage of him.In short, the originally suggested solution (player 1 to playstrategy 1 and player 2 to play strategy 3) is an unstablesolution, so it is necessary to develop a more satisfactory

solution. But what kind of solution should it be?


19/23

The key fact seems to be that whenever one players strategy ispredictable, the opponent can take Advantage of thisinformation to improve his position. Therefore, an essential feature of a rational plan for playing a

game such as this one is that neither player should be ableto deduce which strategy the other will use. Hence, in thiscase, rather than applying some known criterion fordetermining a single strategy that will definitely be used, it isnecessary to choose among alternative acceptable strategieson some kind of random basis.

By doing this, neither player knows in advance which of hisown strategies will be used, let alone what his opponent willdo.


20/23

When game is played pure strategies are chosen by using random device to obtain arandom observation from the probability.

These plans (x1, x2, . . . , xm) and ( y1, y2, . . . , yn) are usually referred to asmixed strategies, and the original strategies are then called pure strategies

To play the game each player could then flip a coin to determine which of his twoacceptable pure strategies he will actually use


21/23

In this context, the minimax criterion says that a given player should

select the mixed strategy that minimizes the maximum expected loss tohimself.

This criterion says to maximin instead, i.e., maximize the minimum expected payoffto the player. By the minimum expected payoff we mean the smallest possibleexpected payoff that can result from any mixed strategy with which the opponent cancounter. Thus, the mixed strategy for player 1 that is optimal according to this criterion

is the one that provides the guarantee (minimum expected payoff) that is best(maximal). (The value of this best guarantee is the maximin value, denoted by v.)Similarly, the optimal strategy for player 2 is the one that provides the best guarantee,where best now means minimal and guarantee refers to the maximum expected lossthat can be administered by any of the opponents mixed strategies. (This bestguarantee is the minimax value, denoted by v


22/23

games not having a saddle point turned out to be unstable (no stablesolutions)


23/23

games not having a saddle point turned out to be unstable (no stablesolutions)

Documents

Game Theory - Chapter No 14 - Presentation