Statistics 5802

Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for Windows software Modeling

A model of reality Elements Players Rules Strategies Payoffs

Players - each player is an individual or group of individuals with similar interests (corporation, nation, team) Single player game – game against nature decision table

To what extent can the players communicate with one another? Can the players enter into binding agreements? Can rewards be shared? What information is available to each player? Tic-tac-toe vs. let’s make a deal Are moves sequential or simultaneous?

Strategies - a complete specification of what to do in all situations strategy versus move Examples – tic tac toe; let's make a deal

Causal relationships - players' strategies lead to outcomes/payoffs Outcomes are based on strategies of all players Outcomes are typically $ or utils long run Payoff sums 0 (poker, tic-tac-toe, market share change) Constant (total market share) General (let’s make a deal) Payoff representation For many games if there are n-players the outcome is represented by a list of n payoffs. Example – market share of 4 competing companies - (23,52,8,7)

Number of players 1, 2 or more than 2 Total reward zero sum or constant sum vs non zero sum Information perfect information (everything known to every player) or not chess and checkers - games of perfect information bridge, poker - not games of perfect information

Is there a "solution" to the game? Does the concept of a solution exist? Is the concept of a solution unique? What should each player do? (What are the optimal strategies?) What should be the outcome of the game? (e.g.-tic tac toe – tie; ) What is the power of each player? (stock holders, states, voting blocs) What do (not should) people do (experimental, behavioral)

Table – generally for simultaneous moves Tree – generally for sequential moves

A woman (Ellen) and her husband (Pat) each have two choices for entertainment on a particular Saturday night. Each can either go to a WWE match or to a ballet. Ellen prefers the WWE match while Pat prefers the ballet. However, to both it is more important that they go out together than that they see the preferred entertainment.

Ellen\PatWWEBallet WWE(2, 1)(-1, -1) Ballet(-1, -1)(1, 2)

Ellen\PatWWEBallet WWE(2, 1)(-1, -1) Ballet(-1, -1)(1, 2) Do players see the same reward structure? (assume yes) Are decisions made simultaneously or does one player go first? (If one player goes first a tree is a better representation) Is communication permitted? Is game played once, repeated a known number of times or repeated an “infinite” number of times.

 Determine what Pat would do at each of the Pat nodes … Compare 1 and -1 Compare -1 and 2

… then determine what Ellen should do Compare 1 and -1 Compare - 1 and 2 Compare 2 and 1

In a game such as the Battle of the Sexes a preemptive decision will win the game for you!!

2 players Opposite interests (zero sum) communication does not matter binding agreements do not make sense

Row has m strategies Column has n strategies Row and column select a strategy simultaneously The outcome (payoff to each player) is a function of the strategy selected by row and the strategy by column The sum of the payoffs is zero

Column pays row the amount in the cell Negative numbers mean row pays column

Row collects some amount between 14 and 67 from column in this game Decisions are simultaneous Note: The game is unfair because column can not win. Ultimately, we want to find out exactly how unfair this game is

Rows, columns or both can be interchanged without changing the structure of the game. In the two games below Rows 1 and 2 have been interchanged but the games are identical

Reminder: Column pays row the amount in the chosen cell. You are row. Should you select row 1 or row 2 and why? Remember, row and column select simultaneously.

Reminder: Column pays row the amount in the chosen cell. You are column. Should you select col 1 or col 2 and why? Remember, row and column select simultaneously.

Reminder: Column pays row the amount in the chosen cell. We say that row 2 dominates row 1 since each outcome in row 2 is better than the corresponding outcome in row 1 Similarly, we say that column 1 dominates column 2 since each outcome in column 1 is better than the corresponding outcome in column 2.

We can always eliminate rows or columns which are dominated in a zero sum game.

Reminder: Column pays row the amount in the chosen cell. Thus, we have solved our first game (and without using QM for Windows.) Row will select row 2, Column will select col 1 and column will pay row $34. We say the value of the game is $34. We previously had said that this game is unfair because row always wins. To make the game fair, row should pay column $34 for the opportunity to play this game.

 Game  Splitting a piece of cake  In two  Statistician  Game theorist  In more than two  Team work division  Splitting work for projects

Answer the following 3 questions before going to the following slides. What should row do? (easy question) What should column do? (not quite as easy) What is the value of the game (easy if you got the other 2 questions)

As was the case before, row should select row 2 because it is better than row 1 regardless of which column is chosen. That is, $55 is better than $18 and $30 is better than $24.

Until now, we have found that one row or one column dominates another. At this point though we have a problem because there is no column domination. $18 < $24 But $55 > $30 Therefore, neither column dominates the other.

However, when column examines this game, column knows that row is going to select row 2. Therefore, column’s only real choice is between paying $55 and paying $30. Column will select col 2, and lose $30 to row in this game. Notice the “you know, I know” logic.

Answer the following 3 questions before going to the following slides. What should row do? (difficult question) What should column do? (difficult question) What is the value of the game (doubly difficult question since the first two questions are difficult)

This game has no dominant row nor does it have a dominant column. Thus, we have no straightforward answer to this problem.

Row could take the following conservative (maximin) approach to this problem. Row could look at the worst that can happen in either row. That is, if row selects row 1, row may end up winning only $25 whereas if row selects row 2 row may end up winning only $14. Therefore, row prefers row 1 because the worst case ($25) is better than the worst case ($14) for row 2.

Since $25 is the best of the worst or maximum of the minima it is called the maximin. This is the same analysis as if row goes first. Note: It is disadvantageous to go first in a zero sum game.

Column could take a similar conservative (minimax) approach. Column could look at the worst that can happen in either column. That is, if column selects col 1, column may end up paying as much as $34 whereas if column selects col 2 column may end up paying as much as $67. Therefore, column prefers col 1 because the worst case ($34) is better than the worst case ($67) for column 2.

Since $34 is the best of the worst or minimum of the maxima for column it is called the minimax. This is the same analysis as if column goes first. Note: It is disadvantageous to go first in a zero sum game.

When we put row and column’s conservative approaches together we see that row will play row 1, column will play column 1 and the outcome (value) of the game will be that column will pay row $25 (the outcome in row 1, column 1). What is wrong with this outcome?

If row knows that column will select column 1 because column is conservative then row needs to select row 2 and get $34 instead of $25.

However, if column knows that row will select row 2 because row knows that column is conservative then column needs to select col 2 and pay only $14 instead of $34.

However, if row knows that column knows that row will select row 2 because row knows that column is conservative and therefore column needs to select col 2 then row must select row 1 and collect $67 instead of $14.

However, if column knows that row knows that column knows that row will select row 2 because row knows that column is conservative and therefore column needs to select col 2 and that therefore row must select row 1 then column must select col 1 and pay $25 instead of $67 and we are back where we began.

The structure of this game is different from the structure of the first two examples. They each had only one entry as a solution and in this game we keep cycling around. There is a lesson for this game ….

The only way to not let your opponent take advantage of your choice is to not know what your choice is yourself!!! That is, you must select your strategy randomly. We call this a mixed strategy.

You must select your strategy randomly!!!

Notice that in examples 1 & 2 (which are trivial to solve) we have that maximin = minimax maximin Minimax

Notice that in game 3 (which is hard to solve) we have that maximin < minimax. The Value of the game is between maximin, minimax maximin Minimax

Row will pick row 1 with probability p and row 2 with probability (1-p) For now, ignore the fact that column also should mix strategies

How will column respond to any value of p for row?

We need to find p to maximize the minimum expected value against every column We need to find q to minimize the maximum expected value against every row

Row should play row 1 32% of the time and row 2 68% of the time. Column should play column 1 85% of the time and column 2 15% of the time. On average, column will pay row $31.10.

If row and column each play according to the percentages on the outside then each of the four cells will occur with probabilities as shown in the table

This leads to an expected value of 25* * * *.098 =

If maximin=minimax there is a saddle point (equilibrium) and each player has a pure strategy – plays only one strategy If maximin does not equal minimax maximin <= value of game <= minimax We find mixed strategies We find the (expected) value or weighted average of the game

 A constant can be added to a zero sum game without affecting the optimal strategies.  A zero sum game can be multiplied by a positive constant without affecting the optimal strategies.  A zero sum game is fair if its value is 0  A graph can be drawn for a player if the player has only 2 strategies available.

Models (see Word document)