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Unformatted text preview: Lecture Slides Week #1
L
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.
p y
– There is some choice of action where strategy matters.
– The game has one or more outcomes, e.g. someone wins,
someone loses.
– The outcome depends on the strategies chosen by all players;
there is strategic interaction.
interaction
• What does this rule out?
– Games of pure chance, e.g. lotteries, slot machines. (Strategies
chance e g lotteries
machines
don't matter).
– Games without strategic interaction between players, e.g.
g
p y
g
Solitaire Why Do Economists Study Games?
y
y
• Games are a convenient way in which to model
the strategic interactions among economic agents.
• Many economic issues involve strategic
interaction.
interaction
– Behavior in imperfectly competitive markets, e.g.
CocaCola versus Pepsi.
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
negotiations.
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
each player
player.
3. A complete description of the information
available to players at each decision node.
il bl
l
hd i i
d
p
q
(p y ff )
4. A description of the consequences (payoffs)
for each player for every possible profile of
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p y
strategy choices of all players. The Prisoners Dilemma Game
Prisoners'
• Two players, prisoners 1, 2. There is no physical evidence to
convict either one, so the prosecutor seeks a confession.
one
confession
• Each prisoner has two strategies.
– Pi
Prisoner 1: Don't C f
1 D 't Confess, C f
Confess
– Prisoner 2: Don't Confess, Confess
– P ff consequences are quantified i prison years.
Payoff
tifi d in i
• More years= worse payoffs. – Prisoner 1 payoff first followed by professor 2 payoff
first,
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
Pi
’ Dilemma in “Normal” or
i “N
l”
“Strategic” Form
Prisoner 2↓
Don't
D 't
Confess Confess Don't
Confess
C f 1,1 15,0 Co ess
Confess 0, 5
0,15 5,5
5, 5 Prisoner 1↓ • Think of the payoffs as prison terms/years lost
p y
p
y How to play games using the
comlabgames software.
• Double click on Comlabgames desktop icon.
• Cli k on ‘Client Play’ tab.
Click ‘Cli
l ’ b
• Replace “localhost” with this address:
136.142.72.19:9876
do)
• Enter a user name and password (any will do).
Then click the login button.
• Start playing when your role is assigned
assigned.
• You are randomly matched with one other player.
• Choose a row or column depending on your role. Computer Screen View
C
S
i Results S
R lt Screen View
Vi
Number of times
each outcome has
o tcome
been realized. Number of
times each outcome
has been played
h b
l d Prisoners'' Dil
Pi
Dilemma i “Extensive” Form
in “E t i ” F
Prisoner 1
Pi
Don t
Don't
Confess Confess Prisoner 2
Don't
Confess
1,1 This line represents
a constraint on the
information that prisoner
i f
ti th t i
2 has available
(or an “information
set”) While 2 moves
second, h does not
d he d
t
know what 1 has
chosen. Prisoner 2 Confess
15,0 Don't
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
NonZero
• A zerosum game is one in which the players'
interests are in direct conflict, e.g., i football, one
i
i di
fli
in f b ll
team wins and the other loses.
• A game is nonzero 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
p ,
p y
Confess in Prisoners' Dilemma, they both gain
g
relative to both choosing Confess. The Prisoners' Dilemma is
applicable to many other
situations.
i i
• Nuclear arms races.
races
• Efforts to address global warming.
• Dispute Resolution and the decision to hire
a lawyer.
• Corruption/political contributions between
contractors and politicians.
t t
d liti i
• Can you think of other applications?
y
pp Can Communication Help?
C C
i ti H l ?
• Suppose we recognize the Prisoner’s
Prisoner s
Dilemma and we can talk to one another in
advance, for instance, make promises to not
d
f i t
k
i t
t
confess.
• If these promises are nonbinding 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
CheapTalk Collusion in the PD
• Dilbert cartoon
• Golden balls 1
• Golden bal1s 2 Golden Balls i
G ld B ll is not PD
t
• Steal is not a strictly dominant strategy.
• Consider the game in normal form:
g
Player 2
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: N1 others
Jump in
River • What is your strategy? Jump in
River 0,
0 0 1, 5
1 Stay out You Stay
Out 5, 1 10 10 Simultaneous versus Sequential
Move Games
• Games where players choose actions simultaneously
are simultaneous move games.
– Examples: Prisoners Dilemma SealedBid Auctions.
Prisoners' Dilemma,
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.
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
Sender
Don t
Don't
Send If sender sends
(invests) 4, the
amount at stake
is tripled (=12). d
Send 4,0
40 Receiver
Keep
K
0,12 Return
R t
6,6 Computer Screen View • You are either the sender or the receiver. If you
receiver
are the receiver, wait for the sender's decision. One Shot
OneShot versus Repeated Games
• Oneshot: 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.
same players.
– Indefinitely versus finitely repeated games
– Reputational concerns matter; opportunities for
cooperative behavior may arise. • Advise: If you plan to pursue an aggressive strategy
strategy,
ask yourself whether you are in a oneshot or in a
repeated game. If a repeated game think again.
game
game,
again Strategies
• A strategy must be a “comprehensive plan of action”, a decision rule
gy
p
p
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
h
ifi h
i
k i
i i
hi h
ld
conceivably arise during your absence.
• Strategies will depend on whether the game is oneshot or repeated
repeated.
• Examples of oneshot strategies
– Prisoners' Dilemma: Don't Confess, Confess
Don t Confess
– Investment Game:
• Sender: Don t Send, Send
Don't
• Receiver: Keep, Return • How do strategies change when the g
g
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
ll
f h d i
f
i h
i
h
actions chosen in previous plays of the game.
• Most contingent strategies are of the type known as "trigger"
strategies.
• 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.
gy
– 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
information
– 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).
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.
We
l hi
i
ik
b h i All players behave rationally.
– They understand and seek to maximize their own payoffs.
payoffs
– They are flawless in calculating which actions will maximize their
payoffs. Th rules of the game are common k
The l
f h
knowledge:
l d
– Each player knows the set of players, strategies and payoffs from all
p
possible combinations of strategies: call this information “X.”
g
– 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
p y
players
, ,
infinitum. What i C
Wh t is Common Knowledge?
K
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 emailgame. Equilibrium
• The interaction of all (rational) players' strategies
results in an outcome that we call "equilibrium."
l i
h
ll " ilib i
"
• In equilibrium, each player is playing the strategy that
is "b
i a "best response" to the strategies of the other
"
h
i
f h h
players. No one has an incentive to change his
strategy given the strategy choices of the others.
others
• Equilibrium is not:
– Th b t possible outcome. E ilib i
The best
ibl
t
Equilibrium in the oneshot
i th
h t
prisoners' dilemma is for both players to confess.
action
– 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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This note was uploaded on 03/04/2012 for the course ECON 1200 taught by Professor Staff during the Fall '08 term at Pittsburgh.
 Fall '08
 Staff

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