Author Neil Bendle Marketing Metrics Reference Chapter 7 2014 Neil Bendle and Management by the Numbers Inc Game Theory studies competitive and cooperative interactions Despite often using terms such as players and games the ideas apply to markets and managers ID: 681015
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Slide1
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 BendleMarketing Metrics Reference: Chapter 7© 2014 Neil Bendle and Management by the Numbers, Inc.Slide2
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”.
What is Game Theory?2What 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.Slide3
Game Theory and managementGame Theory and Management
3Game 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.InsightGame theory has wide managerial application especially in pricing.
MBTN | Management by the NumbersSlide4
Why study simple Games?
4Why Study Simple Games?
MBTN | Management by the NumbersIn 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.Slide5
The Downside of SimplicityThe Downside of Simplicity
5We
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 NumbersSlide6
Beth
Alfred
Sell Apples
Sell 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
.
Example Game Theory Matrix
Example Game Theory Matrix
6
MBTN | Management by the Numbers
OutcomeSlide7
Example Game Theory MatrixExample Game Theory Matrix
7
BethAlfred
Sell Apples
Sell Oranges
Sell Apples
+2
-2
-2
+2
Sell Oranges
-1
+1
+1
-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 NumbersSlide8
Making The Best ChoiceMaking The Best Choice
8
BethAlfred
Sell Apples
Sell Oranges
Sell
Apples
+2
-2
-2
+2
Sell
Oranges
-1
+1
+1
-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.Slide9
What If manager A Had a Spy?What If Alfred Had a Spy ?
9
BethAlfred
Sell Apples
Sell Oranges
Sell
Apples
+2
-2
-2
+2
Sell Oranges
-1
+1
+1
-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
>
-2Beth will sell oranges.
Alfred can ignore
Beth
’s sell apples column and so Alfred will sell apples as
-1<+2
MBTN | Management by the NumbersSlide10
What If Beth Had a Spy?What If Beth Had a Spy?
10
BethAlfred
Sell Apples
Sell Oranges
Sell Apples
+2
-2
-2
+2
Sell Oranges
-1
+1
+1
-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 NumbersSlide11
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 GamesOr, 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 GamesZero Sum Game or Not?11Zero Sum Game or Not?
MBTN | Management by the NumbersSlide12
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)Zero-Sum GamesDefinitionZero Sum Game = Payoffs in every cell total to zero12Zero-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.Slide13
Example of a Zero-Sum GameExample of a Zero-Sum Game
13
MBTN | Management by the NumbersManager B
Manager A
Action X
Action Y
Action X
+1
-1000
-1
+1000
Action Y
-1000
+1
+1000
-1
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 =0If 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. Slide14
Winning a Zero-Sum Game“Winning” a Zero-Sum Game
14There 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.InsightIn 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 NumbersSlide15
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.Non Zero-Sum Games15Non Zero-Sum GamesMBTN | Management by the Numbers
Definition
Non Zero-Sum Game = Payoffs for one or more cells do not total zero.Slide16
Example of A Non-Zero Sum GameExample of A Non Zero-Sum Game
16
Manager BManager A
Action X
Action Y
Action X
-3
+1
+2
0
Action Y
0
+1
+1
+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 = +1If manager A chooses Y and B
chooses X = +1
+0 = +1If manager
A chooses Y and B chooses Y = +1
+1 = +2
MBTN | Management by the NumbersSlide17
Winning a Non Zero-Sum Game
17Winning a Non Zero-Sum Game
MBTN | Management by the NumbersInsightNon 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.Slide18
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 optionsUse a mixed strategy if the game cannot be solved using dominant optionsSolving Games18Solving GamesMBTN | Management by the NumbersSlide19
A Dominant Action19
A Dominant Action
MBTN | Management by the NumbersDominance 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.Slide20
Example of a Dominant ActionExample of a Dominant Action
20
MBTN | Management by the NumbersAction X is strongly dominant for A as 3>2 and 2>1Q: 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 X
Action Y
Action X
-3
-2
3
2
Action
Y
-2
-1
2
1
Insight
In a situation where one action is dominant, the other actions are referred to as “dominated” actions. Slide21
Solution If manager Has A Dominant Action
21Solution If Manager Has A Dominant Action
MBTN | Management by the NumbersAssume any manager with a dominant action chooses it. Ignore the dominated optionIt 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).Slide22
Solving Games With Two Dominant SolutionsSolving Games With Two Dominant Solutions
22
MBTN | Management by the NumbersAction X is strongly dominant for A as 3>2 and 2>1Action 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 BManager AAction XAction YAction X-3-232
Action Y-2
-1
2
1
SolutionSlide23
Manager B
Manager A
Action X
Action Y
Action X
+2
-1
+2
0
Action Y
0
0
-1
+10
Solving Games With
One Dominant Solution
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.
SolutionSlide24
Mixed StrategiesMixed Strategies
24
MBTN | Management by the NumbersMixed 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.Slide25
What is
A’s best choice? X is better than Y if B
chooses X But Y is better than X if B Chooses YWhat 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.
Identifying Mixed Strategy Games
Identifying Mixed Strategy Games
25
MBTN | Management by the Numbers
Manager B
Manager A
Action X
Action Y
Action X
1
0
3
0
Action Y
0
3
0
1Slide26
Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein,
2nd 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. Game Theory – Further Reference26Game Theory - Further Reference
MBTN | Management by the Numbers