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

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

    Definition: Zero-Sum (Constant-Sum)

    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 2.1.1: 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 given 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 2.1.2: 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 Table \(2.1.1\).

    Table 2.1.1: Payoff Matrix for Cake Cutting Game
    Chooser
    Larger Piece Smaller Piece
    Cutter Cut Evenly (half, half) (half, half)
    Cut Unevenly (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 Table \(2.1.2\).

    Table 2.1.2: Payoff Matrix, in Percent of Cake, for Cake Cutting Game.
    Chooser
    Larger Piece Smaller Piece
    Cutter Cut Evenly \((50, 50)\) \((50, 50)\)
    Cut Uenvenly \((40, 60)\) \((60, 40)\)

    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 Table \(2.1.2\) 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 2.1.3: 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 2.1.4

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

    The solution is given in Table \(2.1.3\).

    Table 2.1.3: Zero-sum Payoff Matrix for Cake Cutting Game.
    Chooser
    Larger Piece Smaller Piece
    Cutter Cut 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 Example \(2.1.4\), it is easy to recognize a zero-sum game since each payoff vector has the form \((a, -a)\) (or \((-a, a)\)).

    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 2.1.5: The Speed Limit Issue

    The candidates have the information given in Table \(2.1.4\) about how they would likely fare in the election based on how they stand on the speed limit.

    Table 2.1.4: Percentage of the Vote for Example \(2.1.5\).
    Bainbridge
    Raise Limit Keep Limit Dodge
    Arnold Raise Limit \((45, 55)\) \((50, 50)\) \((40, 60)\)
    Keep Limit \((60, 40)\) \((55, 45)\) \((50, 50)\)
    Dodge \((45, 55)\) \((55, 45)\) \((40, 60)\)
    Exercise 2.1.1: Analysis of Election Game

    For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.5\).

    1. Explain why Example \(2.1.5\) 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 2.1.6: A New Scenario

    Bainbridge's mother is injured in a highway accident caused by speeding. The new payoff matrix is given in Table \(2.1.6\).

    Table 2.1.5: Percentage of the Vote for Example \(2.1.6\).
    Bainbridge
    Raise Limit Keep Limit Dodge
    Arnold Raise Limit \((45, 55)\) \((10, 90)\) \((40, 60)\)
    Keep Limit \((60, 40)\) \((55, 45)\) \((50, 50)\)
    Dodge \((45, 55)\) \((10, 90)\) \((40, 60)\)
    Exercise 2.1.2: Analysis of the Second Scenario

    For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.6\).

    1. Explain why Example \(2.1.6\) 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 2.1.7: A Third Scenario

    Bainbridge begins giving election speeches at college campuses and monster truck rallies. The new payoff matrix is given in Table \(2.1.6\).

    Table 2.1.6: Percentage of the Vote for Example \(2.1.7\).
    Bainbridge
    Raise Limit Keep Limit Dodge
    Arnold Raise Limit \((35, 65)\) \((10, 90)\) \((60, 40)\)
    Keep Limit \((45, 55)\) \((55, 45)\) \((50, 50)\)
    Dodge \((40, 60)\) \((10, 90)\) \((65, 35)\)
    Exercise 2.1.3: Analysis of the Third Scenario

    For the following questions, assume Arnold and Bainbridge have the payoff matrix given in Example \(2.1.7\).

    1. Explain why Example \(2.1.7\) 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.
    Exercise 2.1.4: Changing the Strategy

    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.

    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: Equilibrium Pair

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

    For example, consider the game matrix from Example \(1.2.1\), Table \(1.2.3\).

    Table \(2.1.7\): Payoff Matrix for Example \(1.2.1\)
    Player 2
    X Y
    Player 1 A \((100, -100)\) \((-10, 10)\)
    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.

    Exercise 2.1.5: Checking Equilibrium Pairs

    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.)

    Exercise 2.1.6: Using "Guess and Check}

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

    1. \(\begin{bmatrix}(2,-2) & (2,-2) \\(1,-1) & (3,-3) \end{bmatrix}\)

  • \(\begin{bmatrix}(3,-3) & (1,-1) \\(2,-2) & (4,-4) \end{bmatrix}\)

  • \(\begin{bmatrix}(4,-4) & (5,-5) \\(4,-4) & (3,-3) \end{bmatrix}\)

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

    Exercise 2.1.7: Existence of Equilibrium Pairs

    Do all games have equilibrium pairs?

    Exercise 2.1.8: Multiple Equilibrium Pairs

    Can a game have more than one equilibrium pair?

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

    Exercise 2.1.9: Rock, Paper, Scissors

    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.

    Exercise 2.1.10: Battle of the Networks

    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. Table \(2.1.8\) gives the payoffs as percent of viewers.

    Table \(2.1.8\): Payoff matrix for Battle of the Networks
    Network 2
    Sitcom Sports
    Network 1 Sitcom \((55, 45)\) \((52, 48)\)
    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?
    Exercise 2.1.11: Competitive Advantage

    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 Table \(2.1.9\). 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.

    Table \(2.1.9\): Payoff matrix for Competitve Advantage
    Firm B
    Adopt New Tech Stay Put
    Firm A Adopt 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.

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

    FAQs

    2.1: Introduction to Two-Person Zero-Sum Games? ›

    Definition: Zero-Sum (Constant-Sum)

    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 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 zero-sum game in operational research? ›

    Zero-Sum Game: Meaning and Definition. 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.

    How do you solve a 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 an example of a zero-sum game? ›

    Zero-sum games are found in many contexts. Poker and gambling are popular examples of zero-sum games since the sum of the amounts won by some players equals the combined losses of the others. Games like chess and tennis, where there is one winner and one loser, are also zero-sum games.

    What are the assumptions of the two-person zero-sum game? ›

    There are two key assumptions about the behavior of the players. The first is that both players are rational. The second is that both players are greedy meaning that they choose their strategies in their own interest (to promote their own wealth).

    How to play zero-sum game? ›

    2 Examples of Zero-Sum Games

    In this game, two players place a penny on a table. If the pennies match (meaning they are both heads or both tails) player A wins the game and keeps both pennies. If the pennies do not match (one is heads and the other is tails), player B wins the game and keeps both pennies.

    What is a two-person zero-sum game with dominance? ›

    A two-person zero-sum game is a type of game theory scenario where the gain of one player is directly balanced by the loss of the other player, and to solve the given game using the dominance property, Player A's strategy B4 dominates B1, and Player B's strategy B3 dominates B1, B2, B4, and B5, resulting in the reduced ...

    What is a two-person game? ›

    A two-player game is a multiplayer game that is played by precisely two players. This is distinct from a solitaire game, which is played by only one player.

    What is a zero-sum game in project management? ›

    Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other.

    Which of the following best describes a zero-sum game? ›

    Mathematicians, economists and analysts use the term zero-sum game throughout game theory and economic theory. It describes the financial gains of one party that cause an equal amount of loss for the other party.

    What is the 2 player 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.

    What is an example of a constant sum game? ›

    Constant-sum games are games of total conflict, which are also called games of pure competition. Poker, for example, is a constant-sum game because the combined wealth of the players remains constant, though its distribution shifts in the course of play.

    What does constant sum mean? ›

    A constant sum scale is a type of question used in a market research survey in which respondents are required to divide a specific number of points or percents as part of a total sum. The allocation of points are divided to detail the variance and weight of each category.

    What is the game that two people can play? ›

    Examples of popular two player games we have are TicTacToe, Master Chess and Basketball Stars.

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