Introduction to Two-Person Zero-Sum Games (2024)

In all of the examples from the last section, whatever one player won, the other player lost.

Definition 2.1.1.

A two player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the terms (or coordinates) in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.

We can always think of zero-sum games as being games in which one player's win is the other player's loss.

Example 2.1.2. Zero-sum in Poker.

Consider a poker game in which each player comes to the game with $100. If there are five players, then the sum of money for all five players is always $500. At any give time during the game, a particular player may have more than $100, but then another player must have less than $100. One player's win is another player's loss.

Example 2.1.3. Zero-sum in Cake Division.

Consider the cake division game. Determine the payoff matrix for this game. It is important to determine what each player's options are first: how can the “cutter” cut the cake? How can the “chooser” pick her piece? The payoff matrix is given in Table2.1.4.

Chooser
Larger PieceSmaller Piece
CutterCut Evenly(half, half)(half, half)
Cut Unvenly(small piece, large piece)(large piece, small piece)

In order to better see that this game is zero-sum (or constant-sum), we could give values for the amount of cake each player gets. For example, half the cake would be 50%, a small piece might be 40%. Then we can rewrite the matrix with the percentage values in Table2.1.5

Chooser
Larger PieceSmaller Piece
CutterCut Evenly\((50, 50)\)\((50, 50)\)
Cut Unvenly\((40, 60)\)\((60, 40)\)

In each outcome, the payoffs to each player add up to 100 (or 100%). In more mathematical terms, the coordinates of each payoff vector add up to 100. Thus the sum is the same, or constant, for each outcome.

It is probably simple to see from the matrix in Table2.1.5 that Player 2 will always choose the large piece, thus Player 1 does best to cut the cake evenly. The outcome of the game is the strategy pair denoted [Cut Evenly, Larger Piece], with resulting payoff vector \((50, 50)\text{.}\)

But why are we going to call these games “zero-sum” rather than “constant-sum”? We can convert any zero-sum game to a game where the payoffs actually sum to zero.

Example 2.1.6. Poker Payoffs Revisited.

Consider the above poker game where each player begins the game with $100. Suppose at some point in the game the five players have the following amounts of money: $50, $200, $140, $100. $10. Then we could think of their gain as -$50, $100, $40, $0, -$90. What do these five numbers add up to?

Example 2.1.7.

Convert the cake division payoffs so that the payoff vectors sum to zero (rather than 100).

The solution is given in Table2.1.8.

Chooser
Larger PieceSmaller Piece
CutterCut Evenly\((0, 0)\)\((0, 0)\)
Cut Unvenly\((-10, 10)\)\((10, -10)\)

But let's make sure we understand what these numbers mean. For example, a payoff of \((0,0)\) does not mean each player gets no cake, it means they don't get any more cake than the other player. In this example, each player gets half the cake (50%) plus the payoff.

In the form of Example2.1.7, it is easy to recognize a zero-sum game since each payoff vector has the form \((a, -a)\) (or \((-a, a)\)).

Subsection 2.1.1 Example: An Election Campaign Game

Two candidates, Arnold and Bainbridge, are facing each other in a state election. They have three choices regarding the issue of the speed limit on I-5: They can support raising the speed limit to 70 MPH, they can support keeping the current speed limit, or they can dodge the issue entirely. The next three examples present three different payoff matrices for Arnold and Bainbridge.

Example 2.1.9. The Speed Limit Issue.

The candidates have the information given in Table2.1.10 about how they would likely fare in the election based on how they stand on the speed limit.

Bainbridge
Raise LimitKeep LimitDodge
Raise Limit\((45, 55)\)\((50, 50)\)\((40, 60)\)
ArnoldKeep Limit\((60, 40)\)\((55, 45)\)\((50, 50)\)
Dodge\((45, 55)\)\((55, 45)\)\((40, 60)\)

For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example2.1.9.

  1. Explain why Example2.1.9 is a zero-sum game.

  2. What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice. And remember, a player doesn't just want to win, he wants to get THE MOST votes– for example, you could assume these are polling numbers and that there is some margin of error, thus a candidate prefers to have a larger margin over his opponent!

  3. What is the outcome of the election?

  4. Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.

Example 2.1.12. A New Scenario.

Bainbridge's mother is injured in a highway accident caused by speeding. The new payoff matrix is given in Table2.1.13.

Bainbridge
Raise LimitKeep LimitDodge
Raise Limit\((45, 55)\)\((10, 90)\)\((40, 60)\)
ArnoldKeep Limit\((60, 40)\)\((55, 45)\)\((50, 50)\)
Dodge\((45, 55)\)\((10, 90)\)\((40, 60)\)

For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example2.1.12.

.
  1. Explain why Example2.1.12 is a zero-sum game.

  2. What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice.

  3. What is the outcome of the election?

  4. Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.

Example 2.1.15. A Third Scenario.

Bainbridge begins giving election speeches at college campuses and monster truck rallies. The new payoff matrix is given in Table2.1.16.

Bainbridge
Raise LimitKeep LimitDodge
Raise Limit\((35, 65)\)\((10, 90)\)\((60, 40)\)
ArnoldKeep Limit\((45, 55)\)\((55, 45)\)\((50, 50)\)
Dodge\((40, 60)\)\((10, 90)\)\((65, 35)\)

For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example2.1.15.

  1. Explain why Example2.1.15 is a zero-sum game.

  2. What should Arnold choose to do? What should Bainbridge choose to do? Be sure to explain each candidate's choice.

  3. What is the outcome of the election?

  4. Does Arnold need to consider Bainbridge's strategies is in order to decide on his own strategy? Does Bainbridge need to consider Arnold's strategies is in order to decide on his own strategy? Explain your answer.

In each of the above scenarios, is there any reason for Arnold or Bainbridge to change his strategy? If there is, explain under what circ*mstances it makes sense to change strategy. If not, explain why it never makes sense to change strategy.

Subsection 2.1.2 Equilibrium Pairs

Chances are, in each of the exercises above, you were able to determine what each player should do. In particular, if both players play your suggested strategies, there is no reason for either player to change to a different strategy.

Definition 2.1.19.

A pair of strategies is an equilibrium pair if neither player gains by changing strategies.

For example, consider the game matrix from Example1.2.5, Table1.2.6.

Player 2
XY
Player 1A\((100, -100)\)\((-10, 10)\)
B\((0, 0)\)\((-1, 11)\)

You determined that Player 2 should choose to play Y, and thus, Player 1 should play B (i.e., we have the strategy pair [B, Y]). Why is this an equilibrium pair? If Player 2 plays Y, does Player 1 have any reason to change to strategy A? No, she would lose 10 instead of 1! If Player 1 plays B, does Player 2 have any reason to change to strategy X? No, she would gain 0 instead of 1! Thus neither player benefits from changing strategy, and so we say [B, Y] is an equilibrium pair.

For now, we can use a “guess and check” method for finding equilibrium pairs. Take each outcome and decide whether either player would prefer to switch. Remember, Player 1 can only choose a different row, and Player 2 can only choose a different column. In our above example there are four outcomes to check: [A, X], [A, Y], [B, X], and [B, Y]. We already know [B, Y] is an equilibrium pair, but let's check the rest. Suppose the players play [A, X]. Does Player 1 want to switch to B? No, she'd rather get 100 than 0. Does player 2 want to switch to Y? Yes! She'd rather get 10 than -100. So [A, X] is NOT an equilibrium pair since a player wants to switch. Now check that for [A, Y] Player 1 would want to switch, and for [B, X] both players would want to switch. Thus [A, Y] and [B, X] are NOT equilibrium pairs. Now you can try to find equilibrium pairs in any matrix game by just checking each payoff vector to see if one of the players would have wanted to switch to a different strategy.

Are the strategy pairs you determined in the three election scenarios equilibrium pairs? In other words, would either player prefer to change strategies? (You don't need to check whether any other strategies are equilibrium pairs.)

Use the “guess and check” method to determine any equilibrium pairs for the following payoff matrices.

  1. \begin{equation*}\left[\begin{matrix}(2,-2)\amp (2, -2)\\(1, -1) \amp (3, -3)\end{matrix}\right]\hspace{.5in}\end{equation*}

  2. \begin{equation*}\left[\begin{matrix}(3,-3)\amp (1, -1)\\(2, -2) \amp (4, -4)\end{matrix}\right]\hspace{.5in}\end{equation*}

  3. \begin{equation*}\left[\begin{matrix}(4,-4)\amp (5, -5)\amp (4, -4)\\(3, -3) \amp (0, 0)\amp (1, -1)\end{matrix}\right]\end{equation*}

After trying the above examples, do you think every game has an equilibrium pair? Can games have multiple equilibrium pairs?

Do all games have equilibrium pairs?

Can a game have more than one equilibrium pair?

The last three exercises give you a few more games to practice with.

Consider the game ROCK, PAPER, SCISSORS (Rock beats Scissors, Scissors beat Paper, Paper beats Rock). Construct the payoff matrix for this game. Does it have an equilibrium pair? Explain your answer.

Two television networks are battling for viewers for 7 pm Monday night. They each need to decide if they are going to show a sitcom or a sporting event. Table2.1.27 gives the payoffs as percent of viewers.

Network 2
SitcomSports
Network 1Sitcom\((55, 45)\)\((52, 48)\)
Sports\((50, 50)\)\((45, 55)\)
  1. Explain why this is a zero-sum game.

  2. Does this game have an equilibrium pair? If so, find it and explain what each network should do.

  3. Convert this game to one in which the payoffs actually sum to zero. Hint: if a network wins 60% of the viewers, how much more than 50% of the viewers does it have?

This game is an example of what economists call Competitive Advantage. Two competing firms need to decide whether or not to adopt a new type of technology. The payoff matrix is in Table2.1.29. The variable \(a\) is a positive number representing the economic advantage a firm will gain if it is the first to adopt the new technology.

Firm A
Adopt New TechStay Put
Firm BAdopt New Tech\((0, 0)\)\((a, -a)\)
Stay Put\((-a, a)\)\((0, 0)\)
  1. Explain the payoff vector for each strategy pair. For example, why should the pair [Adopt New Tech, Stay Put] have the payoff \((a, -a)\text{?}\)

  2. Explain what each firm should do.

  3. Give a real life example of Competitive Advantage.

We've seen how to describe a zero-sum game and how to find equilibrium pairs. We've tried to decide what each player's strategy should be. Each player may need to consider the strategy of the other player in order to determine his or her best strategy. But we need to be careful, although our intuition can be useful in deciding the best strategy, we'd like to be able to be more precise about finding strategies for each player. We'll learn some of these tools in the next section.

Introduction to Two-Person Zero-Sum Games (2024)

FAQs

What is the introduction of two person zero-sum game? ›

A two-player game is called a zero-sum game if the sum of the payoffs to each player is constant for all possible outcomes of the game. More specifically, the terms (or coordinates) in each payoff vector must add up to the same value for each payoff vector. Such games are sometimes called constant-sum games instead.

How to solve two person zero-sum game? ›

The 2-person 0-sum game is a basic model in game theory. There are two players, each with an associated set of strategies. While one player aims to maximize her payoff, the other player attempts to take an action to minimize this payoff. In fact, the gain of a player is the loss of another.

What is the payoff matrix for two players? ›

The Payoff Matrix for a simultaneous move game is an array whose rows correspond to the strategies of one player (called the Row player) and whose columns correspond to the strategies of the other player (called the Column player).

What is the saddle point in the two person zero-sum game? ›

The strategies corresponding to the row of maximin value and the column of minimax value are termed the pure optimal strategies for player A and player B, respectively. Also, the position of the element where the row of maximin value and the column of the minimax value intersect is known as the saddle point.

What is the assumption of two person zero-sum game? ›

The behavioural assumption in a zero-sum game is that each player chooses a course of action to maximize the value of his own payoff, while the interests of the two players are completely opposite.

What is the theme of zero-sum game? ›

The zero-sum game definition is: a type of relationship where one participant's loss equates to another participant's gain in a market where the amount of value, such as money and property, is constant. The idea of things such as money and property being fixed is a misunderstanding in economics, however.

What is the solution to the zero sum games? ›

The solution to most zero-sum games lies in Mixed Strategy Equilibria i.e. Choosing from multiple possible strategies using some probability distribution. The only problem is what should be different probabilities with which different actions should be chosen.

How do you beat the zero-sum game? ›

In a zero-sum game, the best strategy is often to try to maximize your own gain while minimizing your opponent's gain.

What is the objective of players in a zero-sum game? ›

A zero-sum game is a scenario that results in a redistribution of a fixed amount, meaning one party gains at another party's expense. A zero-sum game differs from nonzero-sum games, which allow for multiple parties to win or lose simultaneously.

What is optimal strategy in game theory? ›

An optimal strategy is one that provides the best payoff for a player in a game. Optimal Strategy = A strategy that maximizes a player's expected payoff. Games are of two types: cooperative and noncooperative games.

What is dominant strategy in game theory? ›

Dominant strategy game theory refers to a circ*mstance where one player is at a significant advantage in comparison to another player due to the strategy they are employing. Irrespective of what the other player chooses to do, the player using the dominant strategy has full control over the outcome of the game.

What is a pure strategy in game theory? ›

A pure strategy denotes a choice of an available action in games in strategic form. This is a relatively straightforward concept, at least insofar as the notions of availability and actions are well understood. But the concept of strategy also includes pure strategies in extensive games and mixed strategies.

What is the duality gap for two team zero-sum games? ›

The duality gap is the sum of two other quantities, the defensive gaps of the two teams. The defensive gap of Team B is the difference between the payoff to Team B if Team A must play a product strategy, and the payoff to Team B if Team A can use a joint source of randomness (hidden of course from B).

How many winners in a zero-sum game? ›

A zero-sum game is one in which no wealth is created or destroyed. So, in a two-player zero-sum game, whatever one player wins, the other loses.

What are the rules for the zero-sum game? ›

Game Theory

For these games, the sum of the two players' payoffs is always zero; hence, a single number (the amount won by the first player, and therefore lost by the second) determines the payoff.

Who introduced the zero-sum game? ›

Game-theoretic mathematician John von Neumann helped formulate the idea of parties finding equilibrium through a series of two-person, zero-sum games. Along with Oskar Morgenstern, von Neumann wrote a book titled Theory of Games and Economic Behavior (1944), which is the foundational text of the field.

What is a two person constant sum game? ›

What is a Two Person Constant Sum game ? Ans: , The two person games in which the algebraic sum of gains and losses of both the players is a constant is called a two person Constant sum game . In other words a Constant sum Game with only two players is called a two person constant sum game.

What is a two-person game? ›

A two-person game is characterized by the strategies of each player and the payoff matrix. The payoff matrix shows the gain (positive or negative) for player 1 that would result from each combination of strategies for the two players.

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