Unformatted text preview: Lecture Slides Week #1
Slid W k
Game Theory Concepts W a s Ga e?
What is a Game?
• There are many types of games, board games, card games, video
games, field games (e.g. football), etc.
• We focus on games where:
– There are 2 or more players.
– There is some choice of action where strategy matters.
– The game has one or more outcomes, e.g. someone wins,
– The outcome depends on the strategies chosen by all players;
there is strategic interaction.
• What does this rule out?
– Games of pure chance, e.g. lotteries, slot machines. (Strategies
chance e g lotteries
– Games without strategic interaction between players, e.g.
Solitaire Why Do Economists Study Games?
• Games are a convenient way in which to model
the strategic interactions among economic agents.
• Many economic issues involve strategic
– Behavior in imperfectly competitive markets, e.g.
Coca-Cola versus Pepsi.
– Behavior in auctions, e.g., bidders bidding against
other bidders who have private valuations for the item.
– Behavior in economic negotiations, e.g. trade
g • Game theory is not limited to economics!! Four Elements of a Game:
1. The players
– how many players are there?
– does nature/chance play a role?
2. A complete description of the strategies of
3. A complete description of the information
available to players at each decision node.
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(p y ff )
4. A description of the consequences (payoffs)
for each player for every possible profile of
strategy choices of all players. The Prisoners Dilemma Game
• Two players, prisoners 1, 2. There is no physical evidence to
convict either one, so the prosecutor seeks a confession.
• Each prisoner has two strategies.
Prisoner 1: Don't C f
1 D 't Confess, C f
– Prisoner 2: Don't Confess, Confess
– P ff consequences are quantified i prison years.
tifi d in i
• More years= worse payoffs. – Prisoner 1 payoff first followed by professor 2 payoff
• Information about strategies and payoffs is complete; both
players (prisoners) know the available strategies and the
payoffs from the intersection of all strategies.
• Strategies are chosen by the two Prisoners simultaneously and
without communication. Prisoners’ Dil
’ Dilemma in “Normal” or
Confess Confess Don't
C f -1,-1 -15,0 Co ess
Confess 0, 5
5, 5 Prisoner 1↓ • Think of the payoffs as prison terms/years lost
y How to play games using the
• Double click on Comlabgames desktop icon.
• Cli k on ‘Client Play’ tab.
l ’ b
• Replace “localhost” with this address:
• Enter a user name and password (any will do).
Then click the login button.
• Start playing when your role is assigned
• You are randomly matched with one other player.
• Choose a row or column depending on your role. Computer Screen View
i Results S
R lt Screen View
Number of times
each outcome has
been realized. Number of
times each outcome
has been played
l d Prisoners'' Dil
Dilemma i “Extensive” Form
in “E t i ” F
Confess Confess Prisoner 2
1,1 This line represents
a constraint on the
information that prisoner
ti th t i
2 has available
(or an “information
set”) While 2 moves
second, h does not
d he d
know what 1 has
chosen. Prisoner 2 Confess
0,15 Payoffs are: Prisoner 1 payoff, Prisoner 2 payoff. Confess
5,5 Computer Screen View Prisoners
Prisoners' Dilemma is an example
of a Non Zero Sum Game
• A zero-sum game is one in which the players'
interests are in direct conflict, e.g., i football, one
in f b ll
team wins and the other loses.
• A game is non-zero sum, if players’ interests are not
always in direct conflict, so that there are
opportunities for both to gain.
• For example, when both players choose Don't
Confess in Prisoners' Dilemma, they both gain
relative to both choosing Confess. The Prisoners' Dilemma is
applicable to many other
• Nuclear arms races.
• Efforts to address global warming.
• Dispute Resolution and the decision to hire
• Corruption/political contributions between
contractors and politicians.
d liti i
• Can you think of other applications?
pp Can Communication Help?
i ti H l ?
• Suppose we recognize the Prisoner’s
Dilemma and we can talk to one another in
advance, for instance, make promises to not
f i t
• If these promises are non-binding and / or
there are little consequences from breaking
these promises (they are “cheap talk”) then
the ability of the prisoners to communicate
prior to choosing their strategies may not
matter. Illustration of Problems with
Cheap-Talk Collusion in the PD
• Dilbert cartoon
• Golden balls 1
• Golden bal1s 2 Golden Balls i
G ld B ll is not PD
• Steal is not a strictly dominant strategy.
• Consider the game in normal form:
Split Steal Player Split 50%, 50% 0%, 100% 1 Steal 100%, 0% 0%, 0% • If you think your opponent will steal, you are
indifferent between stealing and splitting. Why? In
that case, both strategies yield the same payoff, 0%. The Volunteer’s Dilemma:
also has no dominant strategy
• • A group of N people including you are standing on the riverbank and observe
that a stranger is drowning in the treacherous river. Do you jump in to save the
pe so or stay out?
person o s y ou ?
Suppose the game can be be assigned payoffs as follows: N-1 others
River • What is your strategy? Jump in
0 0 -1, 5
1 Stay out You Stay
Out 5, -1 -10 -10 Simultaneous versus Sequential
• Games where players choose actions simultaneously
are simultaneous move games.
– Examples: Prisoners Dilemma Sealed-Bid Auctions.
– Must anticipate what your opponent will do right now,
recognizing that your opponent is doing the same. • Games where players choose actions in a particular
sequence are sequential move games.
– Examples: Chess, Bargaining/Negotiations.
– Must look ahead in order to know what action to choose
now. • Many strategic situations involve both sequential and
simultaneous moves. The Investment Game is a
Sequential Move Game
Send If sender sends
(invests) 4, the
amount at stake
is tripled (=12). d
6,6 Computer Screen View • You are either the sender or the receiver. If you
are the receiver, wait for the sender's decision. One Shot
One-Shot versus Repeated Games
• One-shot: play of the game occurs once.
– Players likely to not know much about one another.
– Example - tipping on your vacation • Repeated: play of the game is repeated with the
sa e p aye s.
– Indefinitely versus finitely repeated games
– Reputational concerns matter; opportunities for
cooperative behavior may arise. • Advise: If you plan to pursue an aggressive strategy
ask yourself whether you are in a one-shot or in a
repeated game. If a repeated game think again.
• A strategy must be a “comprehensive plan of action”, a decision rule
or set of instructions about which actions a player should take
• It is the equivalent of a memo, left behind when you go on vacation,
that specifies the actions you want taken in every situation which could
conceivably arise during your absence.
• Strategies will depend on whether the game is one-shot or repeated
• Examples of one-shot strategies
– Prisoners' Dilemma: Don't Confess, Confess
Don t Confess
– Investment Game:
• Sender: Don t Send, Send
• Receiver: Keep, Return • How do strategies change when the g
game is repeated?
p Repeated Game Strategies
• In repeated games, the sequential nature of the relationship
allows for the adoption of strategies that are contingent on the
f h d i
actions chosen in previous plays of the game.
• Most contingent strategies are of the type known as "trigger"
• Example trigger strategies
– In prisoners' dilemma: Initially play Don't confess. If your opponent
plays Confess, then play Confess in the next round. If your opponent
plays Don't confess, then play Don't confess in the next round. This is
known as the "tit for tat" strategy.
– In the investment game, if you are the sender: Initially play Send. Play
Send as long as the receiver plays Return. If the receiver plays Keep,
never play Send again. This is known as the "grim trigger" strategy. Information
• Players have perfect information if they know
exactly what has happened every time a
decision needs to be made, e.g. in Chess.
• Otherwise, the game is one of imperfect
– Example: In the repeated investment game, the
sender and receiver might be differentially
informed about the investment outcome. For
example, the receiver may know that the amount
invested is always tripled, but the sender may not
be aware of this fact. Assumptions Game Theorists Make Payoffs are known and fixed. People treat expected payoffs
the same as certain payoffs (they are risk neutral).
– Example: a risk neutral person is indifferent between $25 for certain or
a 25% chance of earning $100 and a 75% chance of earning 0.
– W can relax this assumption to capture risk averse behavior.
b h i All players behave rationally.
– They understand and seek to maximize their own payoffs.
– They are flawless in calculating which actions will maximize their
payoffs. Th rules of the game are common k
– Each player knows the set of players, strategies and payoffs from all
possible combinations of strategies: call this information “X.”
– Common knowledge means that each player knows that all players
know X, that all players know that all players know X, that all players
know that all players know that all p y know X and so on,..., ad
infinitum. What i C
Wh t is Common Knowledge?
l d ?
• Common knowledge means that everyone knows that everyone knows
that everyone knows….
• Things that might be regarded as common knowledge:
– Right/left hand side of the road
– There are 7 days in a week. • Things that may not be regarded as common knowledge:
– Amount of fish caught by Philippine fishermen in 2010?
• [290,000 metric tons] – The capital of Botswana?
• [Gaborone] – Henry the VIII’s third wife.
• [Jane Seymour] • Uncertainty of communication can mean a lack of comon knowledge,
e.g. the email-game. Equilibrium
• The interaction of all (rational) players' strategies
results in an outcome that we call "equilibrium."
ll " ilib i
• In equilibrium, each player is playing the strategy that
i a "best response" to the strategies of the other
f h h
players. No one has an incentive to change his
strategy given the strategy choices of the others.
• Equilibrium is not:
– Th b t possible outcome. E ilib i
Equilibrium in the one-shot
prisoners' dilemma is for both players to confess.
– A situation where players always choose the same action.
Sometimes equilibrium will involve changing action
choices (known as a mixed strategy equilibrium). ...
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- Fall '08
- Game Theory, players, don t, confess