Examples+of+Games

=**Main features of Games**=

One way to describe a game is by listing the players (or individuals) participating in the game, and for each player, listing the alternative choices (called actions or strategies) available to that player. In the case of a two-player game, the actions of the first player form the rows, and the actions of the second player the columns, of a matrix. The entries in the matrix are two numbers representing the utility or payoff to the first and second player respectively. A very famous game is the Prisoner's Dilemma game. In this game the two players are partners in a crime who have been captured by the police. Each suspect is placed in a separate cell, and offered the opportunity to confess to the crime. The game can be represented by the following matrix of payoffs
 * || not confess || confess ||
 * not confess || 5,5 || -4,10 ||
 * confess || 10,-4 || 1,1 ||

Note that higher numbers are better (more utility). If neither suspect confesses, they go free, and split the proceeds of their crime which we represent by 5 units of utility for each suspect. However, if one prisoner confesses and the other does not, the prisoner who confesses testifies against the other in exchange for going free and gets the entire 10 units of utility, while the prisoner who did not confess goes to prison and which results in the low utility of -4. If both prisoners confess, then both are given a reduced term, but both are convicted, which we represent by giving each 1 unit of utility: better than having the other prisoner confess, but not so good as going free. This game has fascinated game theorists for a variety of reasons. **First**, it is a simple representation of a variety of important situations. For example, instead of confess/not confess we could label the strategies "contribute to the common good" or "behave selfishly." This captures a variety of situations economists describe as public goods problems. An example is the construction of a bridge. It is best for everyone if the bridge is built, but best for each individual if someone else builds the bridge. This is sometimes refered to in economics as an externality. Similarly this game could describe the alternative of two firms competing in the same market, and instead of confess/not confess we could label the strategies "set a high price" and "set a low price." Naturally it is best for both firms if they both set high prices, but best for each individual firm to set a low price while the opposition sets a high price. A **second feature** of this game, is that it is self-evident how an intelligent individual should behave. No matter what a suspect believes his partner is going to do, it is always best to confess. If the partner in the other cell is not confessing, it is possible to get 10 instead of 5. If the partner in the other cell is confessing, it is possible to get 1 instead of -4. Yet the pursuit of individually sensible behavior results in each player getting only 1 unit of utility, much less than the 5 units each that they would get if neither confessed. This conflict between the pursuit of individual goals and the common good is at the heart of many game theoretic problems. A **third feature** of this game is that it changes in a very significant way if the game is repeated, or if the players will interact with each other again in the future. Suppose for example that after this game is over, and the suspects either are freed or are released from jail they will commit another crime and the game will be played again. In this case in the first period the suspects may reason that they should not confess because if they do not their partner will not confess in the second game. Strictly speaking, this conclusion is not valid, since in the second game both suspects will confess no matter what happened in the first game. However, repetition opens up the possibility of being rewarded or punished in the future for current behavior, and game theorists have provided a number of theories to explain the obvious intuition that if the game is repeated often enough, the suspects ought to cooperate. Source: http://levine.sscnet.ucla.edu/general/whatis.htm

Popular Game examples
//Coin matching game//
 * Example 1**

Two players choose independently either Head or Tail and report it to a central authority. If both choose the same side of the coin, player 1 wins, otherwise 2 wins. A game has the following :-

<span style="background-color: rgba(255,255,255,0);">1. Set of Players. <span style="background-color: rgba(255,255,255,0);"> The two players who are choosing either Head or Tail in the //Coin Matching Game// form the set of players i.e. P={P1,P2}

<span style="background-color: rgba(255,255,255,0);">2. Set of Rules. <span style="background-color: rgba(255,255,255,0);"> There are ceratin rules which each player has to follow while playing the game. Each player can safely assume that others are following these rules. In coin matching game each player can choose either Head or Tail. He has to act independently and made his selection only once. Player 1 wins if both selections are the same othrwise player 2 wins. These form the Rule set R for the //Coin Matching Game//.

<span style="background-color: rgba(255,255,255,0);">3. Set Strategies Si for each player Pi <span style="background-color: rgba(255,255,255,0);"> For example in Matching coins S1 = { H, T} and S2 = {H,T} are the strategies of the two players. Which means each of them can choose either Head or Tail.

<span style="background-color: rgba(255,255,255,0);">3. Set of Outcomes. O <span style="background-color: rgba(255,255,255,0);"> In matching Coins its {Loss, Win} for both players. <span style="background-color: rgba(255,255,255,0);"> This is a function of the strategy profile selected.

<span style="background-color: rgba(255,255,255,0);">In our example S1 x S2 = {(H,H),(H,T),(T,H),(T,T)} is the strategy profile.

<span style="background-color: rgba(255,255,255,0);"> clearly first and last are win situation for first player while the middle two are win cases for the second player.

<span style="background-color: rgba(255,255,255,0);">4. Pay off ui (o) for each player i and for each outcome o //e// O <span style="background-color: rgba(255,255,255,0);"> This is the amount of benifit a player derives if a particular outcome happens. In general its different for different players.

<span style="background-color: rgba(255,255,255,0);">Let the payoffs in //Coin Matching Game// be, <span style="background-color: rgba(255,255,255,0);">u1(Win) = 100 <span style="background-color: rgba(255,255,255,0);">u1(Loss) = 0 <span style="background-color: rgba(255,255,255,0);">u2(Win) = 100 <span style="background-color: rgba(255,255,255,0);">u2(Loss) = 0

<span style="background-color: rgba(255,255,255,0);">Both the players would like to maximize their payoffs (rationality) so both will try to win. Now lets consider a slightly different case. We redefine the payoffs as,

<span style="background-color: rgba(255,255,255,0);">Player 1 is competetor so <span style="background-color: rgba(255,255,255,0);">u1(Win) = 100 <span style="background-color: rgba(255,255,255,0);">u1(Loss) = 0 <span style="background-color: rgba(255,255,255,0);">While player 2 is a very concerned about seeing player 1 happy (player 1 is his little brother) so for him <span style="background-color: rgba(255,255,255,0);">u2(Win) = 10 <span style="background-color: rgba(255,255,255,0);">u2(Loss) = 100 <span style="background-color: rgba(255,255,255,0);">In this situation only player 1 would try hard to win while player 2 will try to lose. The point to note is that each player tries maximize his payoff for which he/she would like to get the Outcome which gives him maximum payoff.

<span style="background-color: rgba(255,255,255,0);"> Informally we can say the players sit across a table and play the game according to the set of rules. There is an outcome for each player when the game ends. each player derives a pay off from this outcome. For example an outcome of victory brings payoff in terms of awards and fame to the cricket players, while loss means no payoff. Because all the players are rational beings they will try to maximize their payoffs. In non co-operative games players don't know what other players are doing. So they have to make the moves without looking at what others are doing. <span style="background-color: rgba(255,255,255,0);"> Each player chooses a strategy i.e. set of moves he would play.

<span style="background-color: rgba(255,255,255,0);">**Strategy**

<span style="background-color: rgba(255,255,255,0);"> It is the set of moves that a player would play in a game. Being rational a player would chose the startegy in such a way as to maximize his/her payoff. <span style="background-color: rgba(255,255,255,0);">__Zero Sum Game__ : In zero sum game sum of payoff's of all the players for each outcome of the game, is zero. Which means if one player is able to improve his payoff by using some good startegy the payoff of others is going to decrease.

//Prisoner's Dilemma// <span style="background-color: rgba(255,255,255,0);">There are two persons who have committed a crime of which there is no evidence. Police catches them and puts them in two separte cells. Beacuse there is no evidence against the convicts, they cannot be proven guilty. So the police tries to use one againt the other. Each Prisoner is given two options either to //confess// his crime or to //deny// it //.// If prisoner I confesses but prisoner II denies then the first prisoner serves as Testimony against the other and he gets no punishment, while the prisoner II gets full term of 10 yrs and vice versa. If both confess both get 5 years of imprisonment each as now police has evidence against both of them. If both deny the police has evidence against none, so maximum punishment that they can get is 1 yr each. <span style="background-color: rgba(255,255,255,0);">This can be represented in tabular form as.
 * Example 2**

<span style="background-color: rgba(255,255,255,0);">This the standard representation of 2 player game. Each cell has two payoffs, one for each player. The first number in a cell is the penalty of player 1 and the second number is the penalty of player two. Each row represents a startegy for player 1 and each column represents a strategy for player 2. So the bottom right column means if Player 1 denies and Player 2 denies then penalty for player 1 is 1 year and that of player two is also 1 year.
 * <span style="background-color: rgba(255,255,255,0);">I \ II || //<span style="background-color: rgba(255,255,255,0);">Confess // || //<span style="background-color: rgba(255,255,255,0);">Deny // ||
 * //<span style="background-color: rgba(255,255,255,0);">Confess // || <span style="background-color: rgba(255,255,255,0);">5,5 || <span style="background-color: rgba(255,255,255,0);">0,10 ||
 * //<span style="background-color: rgba(255,255,255,0);">Deny // || <span style="background-color: rgba(255,255,255,0);">10,0 || <span style="background-color: rgba(255,255,255,0);">1,1 ||

<span style="background-color: rgba(255,255,255,0);">Now lets analyse the Game with player I 's perspective. <span style="background-color: rgba(255,255,255,0);">He doesn't know if player II is going to confess or deny, but he wants to decrease his punishment. So he considers two cases.

<span style="background-color: rgba(255,255,255,0);">a) If player II confesses <span style="background-color: rgba(255,255,255,0);"> In this case confessing gives 5 years imprisonment while denying gives 10 years <span style="background-color: rgba(255,255,255,0);"> So its better to confess

<span style="background-color: rgba(255,255,255,0);">b) If player II denies <span style="background-color: rgba(255,255,255,0);"> In this case confessing gives only 1 years imprisonment while denying gives 1 years <span style="background-color: rgba(255,255,255,0);"> Again its better to confess <span style="background-color: rgba(255,255,255,0);">So player I will like to confess if he is guilty. <span style="background-color: rgba(255,255,255,0);">Player II will argue on similar lines and will also like to confess if guilty. <span style="background-color: rgba(255,255,255,0);">Lets now assume some numbers to illustrate this fact. If player 1 assumes that player 2 would confess with probability 0.5 .The __expected number of years in prison__ if player one confesses with probability 0.5 i

<span style="background-color: rgba(255,255,255,0);"> 0.5 x 0.5 x ( 5 + 10 + 1 + 0 ) = 4 years. <span style="background-color: rgba(255,255,255,0);">If player I chooses Confess with probability 0.4 and Deny with probability 0.6

<span style="background-color: rgba(255,255,255,0);">He assumes that player II would confess with probability 0.5 <span style="background-color: rgba(255,255,255,0);">for player I <span style="background-color: rgba(255,255,255,0);"> 0.4 x 0.5 x 5 + <span style="background-color: rgba(255,255,255,0);">( I confesses ) ( II confesses ) ( I gets 5 years ) <span style="background-color: rgba(255,255,255,0);"> 0.6 x 0.5 x 10 + <span style="background-color: rgba(255,255,255,0);">( I denies ) ( II confesses ) ( I gets 10 years ) <span style="background-color: rgba(255,255,255,0);"> 0.4 x 0.5 x 0 + <span style="background-color: rgba(255,255,255,0);">( I confesses ) ( II denies ) ( I gets 0 years ) <span style="background-color: rgba(255,255,255,0);"> 0.6 x 0.5 x 1 <span style="background-color: rgba(255,255,255,0);">( I denies ) ( II denies ) ( I gets 1 year ) <span style="background-color: rgba(255,255,255,0);">We see that if he is less likely to confess his penalty increases. __<span style="background-color: rgba(255,255,255,0);">Illustration __ <span style="background-color: rgba(255,255,255,0);">Now we assume <span style="background-color: rgba(255,255,255,0);">Player I confesses with probability q

<span style="background-color: rgba(255,255,255,0);">Player I assumes that player II would confess with probability p <span style="background-color: rgba(255,255,255,0);">for player I <span style="background-color: rgba(255,255,255,0);"> 5 pq + 0 x q(1-p) + 10 x ( 1-q )p + 1.(1-q)(1-p) years <span style="background-color: rgba(255,255,255,0);">= qp - q(4p+1) years <span style="background-color: rgba(255,255,255,0);">this is a decreasing function of q. So more likely player I is to confess less punishment he will get irrespective of what player II does

//Pub managers` Game// In the pub managers game the players are two managers of different village pubs, the King`s Head and the Queen`s Head. Both managers are simultaneously considering introducing a special offer to their customers by cutting the price of their premium beer. Each chooses between making the special offer or not. If one of them makes the offer but the other doesn't the manager who makes the offer will capture some customers from the other and some extra passing trade. But if they both make the offer neither captures customers from the other although they both stand to gain from passing trade. Any increase in customers generates higher revenue for the pub. If neither pub makes the discounted offer the revenue of the Queen`s Head is 7 000 in a week and the revenue to the Kings Head is 8 000. The pay-off matrix for this game is shown in Matrix below which shows the pay-offs as numbers representing revenue per week in thousands of euros.
 * Example 3**

So in this game the pay-offs of the manager of the Queen`s Head are written first and his strategies and pay-offs are highlighted in balk. The matrix shows that if the Queen`s Head manager makes the special offer his pay-off is 10, if the King`s Head manager also makes the offer, and 18 if he doesn't. Similarly if the King`s Head manager makes the offer his pay-off is 14 if the Queen`s Head manager also makes the offer, and 20 if he doesn't.
 * ||  ||= King`s Head ||   ||
 * Queen`s ||  || special offer || no offer ||
 * Head || special offer || **10**, 14 || **18**, 6 ||
 * || no offer || **4**, 20 || **7**, 8 ||

//Cake division//
 * Example 4**

Most people have heard of the repute best way to let two bratty children split a piece of cake. No matter how carefully a parent divides it, one child (or both!) feels he has been slighted with the smaller piece.

The solution is to let one child divide the cake and let the other choose which piece he wants. Greed ensures fair division. The first child can’t object that the cake was divided unevenly because he did it himself. The second child can’t complain since he has his choice of pieces. This homely example is not only a game in von Neumann's sense, but it is also about the simplest illustration of the "minimax" principle upon which game theory is based.

The cake problem is a conflict of interests. Both children want the same thing-as much of the cake as possible. The ultimate division of the cake depends both on how one child cuts the cake and which piece the other child chooses. It is important that each child anticipates what the other will do. This is what makes the situation a game in von Neumann's sense.

Game theory searches for solutions-rational outcomes-of games. Dividing the cake evenly is the best strategy for the first child, since he anticipates that the other child's strategy will be to take the biggest piece. Equal division of the cake is therefore the solution to this game. This solution does not depend on a child's generosity or sense of fair play. It is enforced by both children's self-interest. Game theory seeks solutions of precisely this sort.

Poundstone, William. 1992. Prisoner’s Dilemma. New York: Doubleday.

Further examples and source document: http://cse.iitd.ernet.in/~rahul/cs905/lecture1.html

Nash Equilibrium
In game theory, the Nash equilibrium is a solution 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. Every Game may or may not consist a Nash Equilibrim; There can be more than one Nash Equilibrium. Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. In other words, it provides a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation. Instead, we must ask what each player would do, taking into account the decision-making of the others. Nash equilibrium has been used to analyze hostile situations like war and arms races eg. Prisoner's dilemma.

Source:

https://www.academia.edu/2297327/Game_theory_and_its_business_applications