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Game Theory I This module provides an introduction to game theory for managers and includes the following topics: matrix basics, zero and non-zero sum games, and dominant strategies. Author: Neil Bendle Marketing Metrics Reference: Chapter 7 © 2014 Neil Bendle and Management by the Numbers, Inc.

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Game Theory studies competitive and cooperative interactions. Despite often using terms such as “players” and “games”, the ideas apply to markets and managers. Core idea: Examining incentives allows us to predict what will happen in interactions if people and companies respond “correctly”. W HAT IS G AME T HEORY ? 2 What is Game Theory? MBTN | Management by the Numbers Insight To understand an industry, a manager must comprehend how each participant views the market and their strategic options. In order to do this, managers should consider their partners’ and competitors’ incentives.

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G AME T HEORY AND MANAGEMENT Game Theory and Management 3 Game theory applies to a wide range of managerial situations and choices. “If I drop my price what will my competitor do? How will this impact my profits?” “Should I invest in proprietary technology or license it from a rival?” “Should I enter this new market?” Game theory is also commonly applied to advertising planning, product launches and promotional spending. Insight Game theory has wide managerial application especially in pricing. MBTN | Management by the Numbers

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W HY S TUDY SIMPLE G AMES ? 4 Why Study Simple Games? MBTN | Management by the Numbers In game theory, it is helpful to simplify a complex decision process by focusing on other people’s incentives. We consider simple games: 2 players and 2 possible actions; called a 2x2 game, Single period, Simultaneous (all choose at the same time). Where the payoff covers everything important to a decision maker. Profits are just one example of an objective. Definitions Single Period Game = Choice with no impact on later choices. Payoff = What manager cares about expressed in numbers.

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T HE D OWNSIDE OF S IMPLICITY The Downside of Simplicity 5 We simplify situations to help us clarify our thinking and options, however, the world is not simple and game theory doesn’t capture some important dimensions to strategy. For example: Strategy is Dynamic. In simple game theory, my actions today don’t change the future. Not necessarily the case: My actions affect my competitor’s actions which in turn impact my next decision (pricing, product enhancements, etc.) Considering the future may promote cooperation (i.e. I don’t want to annoy you today if I may need your help tomorrow) Simple game theory assumes no communication between parties. Communication is often incomplete or misleading, but it can impact outcomes. MBTN | Management by the Numbers

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Beth Alfred Sell ApplesSell Oranges Sell Apples Sell Oranges The matrix below describes the managers (e.g. Alfred & Beth) & their possible actions (e.g. Sell Apples or Oranges). The cells represent choice combinations i.e. outcomes. Alfred chooses: Sell Apples Beth chooses: Sell Oranges Read across row for Alfred’s action & down column for Beth’s. E XAMPLE G AME T HEORY M ATRIX Example Game Theory Matrix 6 MBTN | Management by the Numbers Outcome

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E XAMPLE G AME T HEORY M ATRIX Example Game Theory Matrix 7 Beth Alfred Sell ApplesSell Oranges Sell Apples +2-2 +2 Sell Oranges +1 Matrix shows each manager’s payoff for choice combination. Payoffs for each manager in a cell are color coded. e.g. if Alfred chooses to sell apples while manager Beth chooses to sell oranges (highlighted cell in upper right) Alfred gets a payoff of +2. This is his best possible payoff. Beth gets -2 her worst possible payoff. MBTN | Management by the Numbers

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M AKING T HE B EST C HOICE Making The Best Choice 8 Beth Alfred Sell ApplesSell Oranges Sell Apples +2-2 +2 Sell Oranges +1 What is Alfred’s best choice to get the highest payoff? If Beth chooses to sell apples Alfred might get -2 or +1 If Beth chooses to sell oranges Alfred might get +2 or -1 MBTN | Management by the Numbers Insight A person’s best choice often depends on the other player’s choice.

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W HAT I F MANAGER A H AD A S PY ? What If Alfred Had a Spy ? 9 Beth Alfred Sell ApplesSell Oranges Sell Apples +2-2 +2 Sell Oranges +1 The game is easy if Alfred has a spy who says: Beth will sell apples. Alfred can ignore Beth’s sell oranges column and so Alfred will sell oranges as +1>-2 Beth will sell oranges. Alfred can ignore Beth’s sell apples column and so Alfred will sell apples as -1<+2 MBTN | Management by the Numbers

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W HAT I F B ETH H AD A S PY ? What If Beth Had a Spy? 10 Beth Alfred Sell ApplesSell Oranges Sell Apples +2-2 +2 Sell Oranges +1 The game is easy if Beth has a spy who says: Alfred will sell apples. Beth can ignore Alfred’s sell oranges row and so Beth will sell apples as +2>-2 Alfred will sell oranges. Beth can ignore Alfred’s sell apples row and so Beth will sell oranges as -1<+1 MBTN | Management by the Numbers

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The first question to ask is: Are you competing for a fixed pie? The answer to this question falls into two broad categories: Is what I gain offset by an equivalent loss by the other party? (e.g. the size of pie does not change based on my choice because someone else loses the same amount I gain) Fixed Pie = Zero Sum Games Or, is what I gain from my choice different than what the other party loses (e.g. the size of the pie varies based on the choices made by the players) Variable Pie = Non-Zero Sum Games Z ERO S UM G AME OR N OT ? 11 Zero Sum Game or Not? MBTN | Management by the Numbers

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Zero-sum games are common in sport (e.g. darts, football) They are less common in business but do exist (e.g. the award of a big contract) In zero-sum games if one person wins, the other must lose the same amount – the players’ payoffs in any cell total zero (i.e. if manager A gets +1 manager B must get -1) Z ERO -S UM G AMES Definition Zero Sum Game = Payoffs in every cell total to zero 12 Zero-Sum Games MBTN | Management by the Numbers Insight All fixed pie games can easily be recast as zero sum games by subtracting or adding a constant from all payoffs.

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E XAMPLE OF A Z ERO -S UM G AME Example of a Zero-Sum Game 13 MBTN | Management by the Numbers Manager B Manager A Action XAction Y Action X +1-1000 +1000 Action Y -1000+1 +1000 If manager A chooses X and B chooses X: -1 +1 =0 If manager A chooses X and B chooses Y: +1000 -1000 =0 If manager A chooses Y and B chooses X: +1000 -1000 =0 If manager A chooses Y and B chooses Y : -1 +1 =0 The game below is a zero-sum game stacked against B Insight Many zero sum games aren’t “fair”. Payoffs in this example add up to zero in any cell, however Manager A clearly prefers this game.

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W INNING A Z ERO -S UM G AME “Winning” a Zero-Sum Game 14 There is a “best way” to play zero-sum games. Assume the other manager makes their best move and move to minimize the harm they cause you. Your opponent making their best move is a major assumption but this allows you to predict them. Insight In a zero-sum game, assume your competitor will make the best move they can and minimize how much they can harm you. MBTN | Management by the Numbers

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Non zero-sum games are when the payoffs to a given outcome don’t always total zero. The gain by one manager may not be equally offset by the loss by the other manager. In fact, both managers might do better or worse together. Non zero-sum games dominate business. The classic example is a price war. Fighting a price war often destroys the profits of all the firms involved. N ON Z ERO -S UM G AMES 15 Non Zero-Sum Games MBTN | Management by the Numbers Definition Non Zero-Sum Game = Payoffs for one or more cells do not total zero.

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E XAMPLE OF A N ON -Z ERO S UM G AME Example of A Non Zero-Sum Game 16 Manager B Manager A Action XAction Y Action X -3+1 +20 Action Y 0+1 Total payoffs do not equal zero for all the given outcomes If manager A chooses X and B chooses X = +2 -3 = -1 If manager A chooses X and B chooses Y = 0 +1 = +1 If manager A chooses Y and B chooses X = +1 +0 = +1 If manager A chooses Y and B chooses Y = +1 +1 = +2 MBTN | Management by the Numbers

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W INNING A N ON Z ERO -S UM G AME 17 Winning a Non Zero-Sum Game MBTN | Management by the Numbers Insight Non zero-sum games are not simple win/lose situations. In this they mirror most business situations where competing firms often rise or fall at the same time. Winning is hard to define in non zero-sum games. The “correct” way to play sometimes entails managers doing worse than if both were to play “incorrectly”. Non zero-sum games fall into categories with very different outcomes and recommendations for action. We outline these in Game Theory for Managers Part 2.

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That all play the best way they can (optimally) is a major, but useful, assumption. This allows us to determine what should happen in a game. This is sometimes called a "no regrets” solution, i.e. where the person can’t do any better given what the other person does. The solution of a game will be as follows: Look for dominant options, assume dominant options are chosen and ignore dominated options Try to find a solution when ignoring dominated options Use a mixed strategy if the game cannot be solved using dominant options S OLVING G AMES 18 Solving Games MBTN | Management by the Numbers

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A D OMINANT A CTION 19 A Dominant Action MBTN | Management by the Numbers Dominance is when one action is best for a manager regardless of what the other manager does. This makes the game simple: No prediction of the other manager’s action is needed when a dominant action exists. Strong dominance is when one action is always best. Weak dominance is when one action is sometimes better and never worse than the other action. In the case of weak dominance, you should take that action as you will never be worse off and may be better off.

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E XAMPLE OF A D OMINANT A CTION Example of a Dominant Action 20 MBTN | Management by the Numbers Action X is strongly dominant for A as 3>2 and 2>1 Q: Is there a dominant action for Manager B? Action Y is strongly dominant for B as -2>-3 and -1>-2 Manager B Manager A Action XAction Y Action X -3-2 32 Action Y -2 21 Insight In a situation where one action is dominant, the other actions are referred to as “dominated” actions.

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S OLUTION I F MANAGER H AS A D OMINANT A CTION 21 Solution If Manager Has A Dominant Action MBTN | Management by the Numbers Assume any manager with a dominant action chooses it. Ignore the dominated option It is tempting to hope a competitor will take a dominated action if you’d get a big payoff if they did: e.g. “maybe they will do something foolish” In game theory we assume others act correctly, don’t change this assumption however tempting. When an option becomes best only after removing an other player’s option this is called iterated dominance. (This does not count as dominance in the initial matrix).

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S OLVING G AMES W ITH T WO D OMINANT S OLUTIONS Solving Games With Two Dominant Solutions 22 MBTN | Management by the Numbers Action X is strongly dominant for A as 3>2 and 2>1 Action Y is strongly dominant for B as -2>-3 and -1>-2 So ignore action Y for A and action X for B A will choose X and B Y. Manager A will win 2 and B will lose 2 each time they play this game. (B hates this game) Manager B Manager A Action XAction Y Action X -3-2 32 Action Y -2 21 Solution

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Manager B Manager A Action XAction Y Action X +2 +20 Action Y 00 +10 S OLVING G AMES W ITH O NE D OMINANT S OLUTION 23 Solving Games With One Dominant Solution MBTN | Management by the Numbers What should manager A do? A’s best choice isn’t clear But manager A knows B won’t choose Y as it is weakly dominated by X +2>1 & 0=0 Ignore manager B choosing Y Given manager B will choose X manager A should choose action X, +2>-1 The solution is that both choose X. Note although both choosing Y gives A a tempting payoff of 10 it simply won’t happen and should be ignored. Solution

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M IXED S TRATEGIES Mixed Strategies 24 MBTN | Management by the Numbers Mixed strategies involve a manager being deliberately unpredictable, using a random strategy. Sometimes tax authorities audit, sometimes they don’t. Mixed strategies are useful when predictability can be taken advantage of, or if everyone using same strategy is ruinous, e.g. think advertising wars when no one backs down. Mixed strategies are when no dominant strategies exist: You find no (strongly or weakly) dominant solutions. Calculating correct mixed strategies is tricky. In this tutorial just identify when a mixed strategy is appropriate.

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What is A’s best choice? X is better than Y if B chooses X But Y is better than X if B Chooses Y What will manager B choose? It isn’t clear either. X is better than Y if A chooses X But Y is better than X if A Chooses Y Assuming the managers cannot discuss their problem the answer is to use a mixed strategy. Both managers do X & Y randomly at calculated probability. I DENTIFYING M IXED S TRATEGY G AMES Identifying Mixed Strategy Games 25 MBTN | Management by the Numbers Manager B Manager A Action XAction Y Action X 10 30 Action Y 03 01

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Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2 nd edition - And - Game Theory for Managers Part 2 Playing the Right Game (advanced MBTN module – Available 2014). This module provides insight into types of games, such as pricing games, that exist and how best to play them. G AME T HEORY – F URTHER R EFERENCE 26 Game Theory - Further Reference MBTN | Management by the Numbers

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