Game Theory 101_ The Complete Textbook ( PDFDrive ) (2)

Game Theory 101: The Complete Textbook
By William Spaniel

Acknowledgements
I thank Varsha Nair for her revisions as I compiled this book. I am also
indebted to Kenny Oyama and Matt Whitten for their further suggestions. I
originally learned game theory from Branislav Slantchev, John Duggan, and
Mark Fey.
Please report possible errors to williamspaniel@gmail.com. I am grateful to
those who have already given me feedback through this medium, and I
encourage you to comment.

About the Author
William Spaniel is a PhD candidate in political science at the University of
Rochester, creator of the popular YouTube series Game Theory 101, and founder
of gametheory101.com. You can email him at williamspaniel@gmail.com or
follow him on Twitter @gametheory101.

Table of Contents
LESSON 1.1: THE PRISONER’S DILEMMA AND STRICT DOMINANCE
LESSON 1.2: ITERATED ELIMINATION OF STRICTLY DOMINATED STRATEGIES
LESSON 1.3: THE STAG HUNT, PURE STRATEGY NASH EQUILIBRIUM, AND BEST
RESPONSES
LESSON 1.4: DOMINANCE AND NASH EQUILIBRIUM
LESSON 1.5: MATCHING PENNIES AND MIXED STRATEGY NASH EQUILIBRIUM
LESSON 1.6: CALCULATING PAYOFFS
LESSON 1.7: STRICT DOMINANCE IN MIXED STRATEGIES
LESSON 1.8: THE ODD RULE AND INFINITELY MANY EQUILIBRIA
LESSON 2.1: GAME TREES AND SUBGAME PERFECT EQUILIBRIUM
LESSON 2.2: BACKWARD INDUCTION
LESSON 2.3: MULTIPLE SUBGAME PERFECT EQUILIBRIA
LESSON 2.4: MAKING THREATS CREDIBLE
LESSON 2.5: COMMITMENT PROBLEMS
LESSON 2.6: BACKWARD INDUCTION WITHOUT A GAME TREE
LESSON 2.7: PROBLEMS WITH BACKWARD INDUCTION
LESSON 2.8: FORWARD INDUCTION
LESSON 3.1: PROBABILITY DISTRIBUTIONS
LESSON 3.2: MIXED STRATEGY NASH EQUILIBRIA IN GENERALIZED GAMES
LESSON 3.3: KNIFE-EDGE EQUILIBRIA
LESSON 3.4: COMPARATIVE STATICS
LESSON 3.5: GENERALIZING MIXED STRATEGY NASH EQUILIBRIUM
LESSON 3.6: ROCK-PAPER-SCISSORS
LESSON 4.1: INFINITE STRATEGY SPACES, SECOND PRICE AUCTIONS, DUELS, AND THE
MEDIAN VOTER THEOREM
MORE FROM WILLIAM SPANIEL

Lesson 1.1: The Prisoner’s Dilemma and Strict
Dominance
At its core, game theory is the study of strategic interdependence—that is,
situations where my actions affect both my welfare and your welfare and vice
versa. Strategic interdependence is tricky, as actors need to anticipate, act, and
react. Blissful ignorance will not cut it.
The prisoner’s dilemma is the oldest and most studied model in game theory,
and its solution concept is also the simplest. As such, we will start with it. Two
thieves plan to rob an electronics store. As they approach the backdoor, the
police arrest them for trespassing. The cops suspect that the pair planned to
break in but lack the evidence to support such an accusation. They therefore
require a confession to charge the suspects with the greater crime.
Having studied game theory in college, the interrogator throws them into the
prisoner’s dilemma. He individually sequesters both robbers and tells each of
them the following:
We are currently charging you with trespassing, which implies a one month
jail sentence. I know you were planning on robbing the store, but right now I
cannot prove it—I need your testimony. In exchange for your cooperation, I will
dismiss your trespassing charge, and your partner will be charged to the fullest
extent of the law: a twelve month jail sentence.
I am offering your partner the same deal. If both of you confess, your
individual testimony is no longer as valuable, and your jail sentence will be
eight months each.
If both criminals are self-interested and only care about minimizing their jail
time, should they take the interrogator’s deal?

1.1.1: Solving the Prisoner’s Dilemma
The story contains a lot of information. Luckily, we can condense everything
we need to know into a simple matrix:
We will use this type of game matrix regularly, so it is important to
understand how to interpret it. There are two players in this game. The first
player’s strategies (keep “quiet” and “confess”) are in the rows, and the second
player’s strategies are in the columns. The first player’s payoffs are listed first
for each outcome, and the second player’s are listed second. For example, if the
first player keeps quiet and the second player confesses, then the game ends in
the top right set of payoffs; the first player receives twelve months of jail time
and the second player receives zero. Finally, as a matter of convention, we refer
to the first player as a man and the second player as a woman; this will allow us
to utilize pronouns like “he” and “she” instead of endlessly repeating “player 1”
and “player 2.”
Which strategy should each player choose? To see the answer, we must look
at each move in isolation. Consider the game from player 1’s perspective.
Suppose he knew player 2 will keep quiet. How should he respond?
Let’s focus on the important information in that context. Since player 1 only
cares about his time in jail, we can block out player 2’s payoffs with question
marks:
Player 1 should confess. If he keeps quiet, he will spend one month in jail.
But if he confesses, he walks away. Since he prefers less jail time to more jail
time, confession produces his best outcome.
Note that player 2’s payoffs are completely irrelevant to player 1’s decision
in this context—if he knows that she will keep quiet, then he only needs to look
at his own payoffs to decide which strategy to pick. Thus, the question marks
could be any number at all, and player 1’s optimal decision given player 2’s
move will remain the same.
On the other hand, suppose player 1 knew that player 2 will confess. What
should he do? Again, the answer is easier to see if we only look at the relevant
information:

Confession wins a second time: confessing leads to eight months of jail time,
whereas silence buys twelve. So player 1 would want to confess if player 2
confesses.
Putting these two pieces of information together, we reach an important
conclusion—player 1 is better off confessing regardless of player 2’s strategy!
Thus, player 1 can effectively ignore whatever he thinks player 2 will do, since
confessing gives him less jail time in either scenario.
Let’s switch over to player 2’s perspective. Suppose she knew that player 1
will keep quiet, even though we realize he should not. Here is her situation:
As before, player 2 should confess, as she will shave a month off her jail
sentence if she does so.
Finally, suppose she knew player 1 will confess. How should she respond?
Unsurprisingly, she should confess and spend four fewer months in jail.
Once more, player 2 prefers confessing regardless of what player 1 does.
Thus, we have reached a solution: both players confess, and both players spend
eight months in jail. The justice system has triumphed, thanks to the
interrogator’s savviness.
This outcome perplexes a lot of people new to the field of game theory.
Compare the <quiet, quiet> outcome to the <confess, confess> outcome:
Looking at the game matrix, people see that the <quiet, quiet> outcome
leaves both players better off than the <confess, confess> outcome. They then
wonder why the players cannot coordinate on keeping quiet. But as we just saw,
promises to remain silent are unsustainable. Player 1 wants player 2 to keep
quiet so when he confesses he walks away free. The same goes for player 2. As a
result, the <quiet, quiet> outcome is inherently unstable. Ultimately, the players

finish in the inferior (but sustainable) <confess, confess> outcome.

1.1.2: The Meaning of the Numbers and the Role of Game Theory
Although a large branch of game theory is devoted to the study of expected
utility, we generally consider each player’s payoffs as a ranking of his most
preferred outcome to his least preferred outcome. In the prisoner’s dilemma, we
assumed that players only wanted to minimize their jail time. Game theory does
not force players to have these preferences, as critics frequently claim. Instead,
game theory analyzes what should happen given what players desire. So if
players only want to minimize jail time, we could use the negative number of
months spent in jail as their payoffs. This preserves their individual orderings
over outcomes, as the most preferred outcome is worth 0, the least preferred
outcome is -12, and everything else logically follows in between.
Interestingly, the cardinal values of the numbers are irrelevant to the
outcome of the prisoner’s dilemma. For example, suppose we changed the
payoff matrix to this:
Here, we have replaced the months of jail time with an ordering of most to
least preferred outcomes, with 4 representing a player’s most preferred outcome
and 1 representing a player’s least preferred outcome. In other words, player 1
would most like to reach the <confess, quiet> outcome, then the <quiet, quiet>
outcome, then the <confess, confess> outcome, then the <quiet, confess>
outcome.
Even with these changes, confess is still always better than keep quiet. To
see this, suppose player 2 kept quiet:
Player 1 should confess, since 4 beats 3.
Likewise, suppose player 2 confessed:
Then player 1 should still confess, as 2 beats 1.
The same is true for player 2. First, suppose player 1 kept quiet:

Player 2 ought to confess, since 4 beats 3.
Alternatively, if player 1 confessed:
Player 2 should confess as well, as 2 is greater than 1. Thus, regardless of
what the other player does, each player’s best strategy is to confess.
To be clear, this preference ordering exclusively over time spent in jail is
just one way the players may interpret the situation. Suppose you and a friend
were actually arrested and the interrogator offered you a similar deal. The results
here do not generally tell you what to do in that situation, unless you and your
friend only cared about jail time. Perhaps your friendship is strong, and both of
you value it more than avoiding jail time. Since confessing might destroy the
friendship, you could prefer to keep quiet if your partner kept quiet, which
changes the ranking of your outcomes. Your preferences here are perfectly
rational. However, we do not yet have the tools to solve the corresponding game.
We will reconsider these alternative sets of preferences in Lesson 1.3.
Indeed, the possibility of alternative preferences highlights game theory’s
role in making predictions about the world. In general, we take a three step
approach: 1) Make assumptions.
2) Do some math.
3) Draw conclusions.
We do steps 1 and 3 everyday. However, absent rigorous logic, some
conclusions we draw may not actually follow from our assumptions. Game
theory—the math from step 2 that this book covers—provides a rigorous way of
ensuring that that our conclusions follow directly from the assumptions. Thus,
correct assumptions imply correct conclusions. But incorrect assumptions could
lead to ridiculous claims. As such, we must be careful (and precise!) about the
assumptions we make, and we should not be surprised if our conclusions change
based on the assumptions we make.
Nevertheless, for the given payoffs in the prisoner’s dilemma, we have seen
an example of strict dominance. We say that a strategy x strictly dominates
strategy y for a player if strategy x provides a greater payoff for that player than
strategy y regardless of what the other players do. In this example, confessing
strictly dominated keeping quiet for both players. Unsurprisingly, players never
optimally select strictly dominated strategies—by definition, a better option
always exists regardless of what the other players do.

1.1.3: Applications of the Prisoner’s Dilemma
The prisoner’s dilemma has a number of applications. Let’s use the game to
explore optimal strategies in a number of different contexts.
First, consider two states considering whether to go to war. The military
technology available to these countries gives the side that strikes first a large
advantage in the fighting. In fact, the first-strike benefit is so great that each
country would prefer attacking the other state even if its rival plays a peaceful
strategy. However, because war destroys property and kills people, both prefer
remaining at peace to simultaneously declaring war.
Using these preferences, we can draw up the following matrix:
From this, we can see that the states most prefer attacking while the other
one plays defensively. (This is due to the first-strike advantage.) Their next best
outcome is to maintain the peace through mutual defensive strategies. After that,
they prefer declaring war simultaneously. Each state’s worst outcome is to
choose defense while the other side acts as the aggressor.
We do not need to solve this game—we already have! This is the same game
from the previous section, except we have exchanged the labels “quiet” with
“defend” and “confess” with “attack.” Thus, we know that both states attack in
this situation even though they both prefer the <defend, defend> outcome. The
first-strike advantages trap the states in a prisoner’s dilemma that leads to war.
A similar problem exists with arms races. Imagine states must
simultaneously choose whether to develop a new military technology.
Constructing weapons is expensive but provides greater security against rival
states. We can draw up another matrix for this scenario:
Here, the states most prefer building while the other state passes. Following
that, they prefer the <pass, pass> outcome to the <build, build> outcome; the
states maintain the same relative military strength in both these outcomes, but
they do not waste money on weaponry if they both pass. The worst possible
outcome is for the other side to build while the original side passes. Again, we
already know the solution to this game. Both sides engage in the arms race and
build.