Application of Game theoryBy Group 3Table of contentGame and
Game theoryWhat is Game?A game is a formal description of a
strategic situation.What is Game theoryGame theory is the formal
study of decision-making where several players must make choices
that potentially affect the interests of the other players.
Types of Game theoryCo-operative vs. Non-co-operativeZero-sum
vs. Non-zero-sumSymmetrical vs. AsymmetricalSimultaneous vs.
SequentialOne-shot vs. Repeated
Co-operative versus Non-co-operative
In game theory, a cooperative game is a game where groups of
players may enforce cooperative behaviour, hence the game is a
competition between coalitions of players, rather than between
individual players.Example: Land owners and workersIn game theory,
a non-cooperative game is one in which players make decisions
independently.Example: Cournot Duopoly model Land-owner and
workers.Let player 1 be a land-owner, and players 2 be workers. The
land-owner needs at least one worker to produce any output.Likewise
any coalition of workers needs the land-owner's participation to
have land to work on, otherwise no production willoccur. We see
that it is natural to model this situation as a coalition game
This game models a situation in which each firm chooses its
output independently, and the market determines the price at which
it is sold. Specifically, if firm1 produces the outputy1and firm2
produces the outputy2then the price at which each unit of output is
sold isP(y1+y2), wherePis the inverse demand function.Denote
firm1's total cost function by TC1(y) and firm2's by TC2(y). Then
firm1's total revenue when the pair of outputs chosen by the firms
is (y1,y2) isP(y1+y2)y1, so that its profit
isP(y1+y2)y1TC1(y1);firm2's revenue isP(y2+y2)y2, and hence its
profit isP(y1+y2)y2TC2(y2).Notice an essential difference between
these specifications of the firms' revenues and those for a
competitive firm or for a monopolist. The revenue of both a
competitive firm and of a monopolist depends only on the
firm'sownoutput: for a competitive firm we assume that the firm's
output does not affect the price, and for a monopolist there are no
other firms in the market. For a duopolist, however, revenue
depends onbothits own outputandthe other firm's output.5Zero-sum
game And Non Zero-sum game
A zero-sum game is a mathematical representation of a situation
in which a participant's gain (or loss) of utility is exactly
balanced by the losses (or gains) of the utility of the other
participant(s).Example: Zero sum gameA nonzero sumdescribes a
situation in which the interacting parties' aggregate gains and
losses are either less than or more than zeroExample:
TradeBXYAX2,-2 3,-3Y-3,30,0Lets say that, due to sea weather
conditions, each island can only send one boat on the first day of
every month, and they will not know if the other island sent a
shipment of food until that boat arrives two days later. So, on the
first day of each month, both islands have two strategies to chose
from: { send, dont send }.Economists sometimespresent these games
using a payoff matrix:
If neither chose to send, they both are stuck with boring
food.3+ 3 = 6If North island sends food and South island does not,
the South will be better off but the North will have lessnomsthan
before.7+ 2 = 9If both chose to send, they will both get to enjoy
delicious crepes with Nutella, resulting in an increase in total
welfare.6 + 6 =126Symmetrical vs. Asymmetrical
A symmetric game is a game where the payoffs for playing a
particular strategy depend only on the other strategies employed,
not on who is playing themExample: Battle of sexesA asymmetric game
is a game where the payoffs for playing a particular strategy
depend on the other strategies employed by both the partiesExample:
TradeWifeMovieCricketHusbandMovie2,10,0Cricket0,01,2A symmetric
game is a game where the payoffs for playing a particular strategy
depend only on the other strategies employed, not on who is playing
them. If the identities of the players can be changed without
changing the payoff to the strategies, then a game is symmetric.
Many of the commonly studied 22 games are symmetric. The standard
representations ofchicken, theprisoner's dilemma, and thestag
huntare all symmetric games. Some[who?]scholars would consider
certain asymmetric games as examples of these games as well.
However, the most common payoffs for each of these games are
symmetric.Most commonly studied asymmetric games are games where
there are not identical strategy sets for both players. For
instance, theultimatum gameand similarly thedictator gamehave
different strategies for each player. It is possible, however, for
a game to have identical strategies for both players, yet be
asymmetric. For example, the game pictured to the right is
asymmetric despite having identical strategy sets for both
players.
7Simultaneous vs. sequential game theoryA simultaneous game is
one in which the players effectively make their decisions at the
same timeexample: rock, paper, scissorA sequential game is one in
which the players take alternate turns to make their
choicesexample: chess
Simultaneous gamesare games where both players move
simultaneously, or if they do not move simultaneously, the later
players are unaware of the earlier players' actions (making
themeffectivelysimultaneous).Sequential games(or dynamic games) are
games where later players have some knowledge about earlier
actions. This need not beperfect informationabout every action of
earlier players; it might be very little knowledge. For instance, a
player may know that an earlier player did not perform one
particular action, while he does not know which of the other
available actions the first player actually performed.The
difference between simultaneous and sequential games is captured in
the different representations discussed above. Often,normal formis
used to represent simultaneous games, andextensive formis used to
represent sequential ones. The transformation of extensive to
normal form is one way, meaning that multiple extensive form games
correspond to the same normal form. Consequently, notions of
equilibrium for simultaneous games are insufficient for reasoning
about sequential games; seesubgame perfection.
8Prisoners DilemmaTwo suspects arrested for a crime. Prisoners
decide whether to confess or not to confessIf both confess, both
sentenced to 3 months of jailIf both do not confess, then both will
be sentenced to 1 month of jailIf one confesses and the other does
not, then the confessor gets freed (0 months of jail) and the
non-confessor sentenced to 9 months of jailWhat should each
prisoner do?
1ConfessNot confess2Confess3,39,0Not confess0,91,11ConfessNot
confess2Confess3,39,0Not confess0,91,1One-shot versus Repeated
In a one-shot game, the stakes are high but carry no further
repercussionsExample: Tipping a waiter on vacation
A repeated game (super game or iterated game) is an extensive
form game which consists in some number of repetitions of some base
gameExample: Passing a bill in ParliamentEconomic applications of
game theoryThe study of oligopolies (industries containing only a
few firms)The study of cartels, e.g., OPECThe study of military
strategiesBargainingAuctions
The study of oligopoliesThe game is based on two premisesEach
player has an incentive to choose an action that benefits itself at
the other players expenseWhen both players act this way, both are
worse of than if they had chosen different actionPlayer AHigh
Production (40 gallons)Low Production (30 gallons)Player BHigh
Production1600,16001500,2000Low Production2000,15001800,1800
13The study of cartelsAcartelis a formal (explicit) "agreement"
amongcompetingfirms OPEC (Open Petroleum Exporting Countries)Case
of Iran and IraqCo-operate with cartel agreementCheating on the
agreement
IranCo-operateCheatIraqCo-operate2,23,1Cheat1,31,1The study of
MilitaryArms Race - Example of possible actions that might have
been taken by U.S. and U.S.S.R
U.S.ArmDisarmUSSRArmAt risk, At riskRisk & weak, safe and
powerfulDisarmSafe & powerful, risk & weakSafe,
SafeBargaining
Co-operative bargainingThe outcomes of negotiations are more
equally beneficial to all members of the householdNon co-operative
bargainingPersonal interests motivate individuals within the
household rather than the desire to work in a collaborative manner
and maximize the benefit of all household membersBargaining
PowerThe relative capacity of each of the parties to a negotiation
or dispute to compel or secure agreements on its own
terms16Bargaining problem terminologiesFeasible setDisagreement
pointEquilibrium Analysis
x: demand from player Ay: demand from player Bd: disagreement
point (often d = 0)
Feasibility set[edit source]Which agreements are feasible
depends on whether bargaining is mediated by an additional party.
When binding contracts are allowed, any joint action is playable,
and the feasibility set consists of all attainable payoffs better
than the disagreement point. When binding contracts are
unavailable, the players can defect (moral hazard), and the
feasibility set is composed of correlated equilibria, since these
outcomes require no exogenous enforcement.Disagreement point[edit
source]The disagreement point v is the value the players can expect
to receive if negotiations break down. This could be some focal
equilibrium that both players could expect to play. This point
directly affects the bargaining solution, however, so it stands to
reason that each player should attempt to choose his disagreement
point in order to maximize his bargaining position. Towards this
objective, it is often advantageous to increase one's own
disagreement payoff while harming the opponent's disagreement
payoff (hence the interpretation of the disagreement as a threat).
If threats are viewed as actions, then one can construct a separate
game wherein each player chooses a threat and receives a payoff
according to the outcome of bargaining. It is known as Nash's
variable threat game. Alternatively, each player could play a
minimax strategy in case of disagreement, choosing to disregard
personal reward in order to hurt the opponent as much as possible
should the opponent leave the bargaining table.Equilibrium
analysis[edit source]
Strategies are represented in the Nash bargaining game by a pair
(x, y). x and y are selected from the interval [d, z], where z is
the total good. If x + y is equal to or less than z, the first
player receives x and the second y. Otherwise both get d. d here
represents the disagreement point or the threat of the game; often
d=0.There are many Nash equilibria in the Nash bargaining game. Any
x and y such that x + y = z is a Nash equilibrium. If either player
increases their demand, both players receive nothing. If either
reduces their demand they will receive less than if they had
demanded x or y. There is also a Nash equilibrium where both
players demand the entire good. Here both players receive nothing,
but neither player can increase their return by unilaterally
changing their strategy.17Nash equilibriumA set of strategies such
that each is best for each player, given that the others are
playing their own equilibrium strategiesNash product = (u(x)
u(d))*(v(y) v(d))u(d) & v(d) : the utility obtained if one
decides not to bargain with the other player
Player AGoStopPlayer BGo-5,-51,0Stop0,1-1,-1
Two person bargaining game
Bargaining solution
Improved disagreement point
Improved disagreement point
AuctionsJan 07, 2009Game Theory23Games of incomplete
informationFirst Price Sealed Bid AuctionBuyers simultaneously
submit their bidsBuyers valuations of the good unknown to each
otherHighest Bidder wins and gets the good at the amount he bid
Second Price Sealed Bid AuctionSame rulesException Winner pays
the second highest bid and gets the good
23
Second-price auctionSuppose you value an item at 100, you should
bid 100 for the itemIf you bid 90Someone bids more than 100: you
lose anywaySomeone bids less than 90: you win anyway and pay
second-priceSomeone bids 95: you lose; you could have won by paying
95If you bid 110Someone bids more than 11o: you lose anywaySomeone
bids less than 100: you win anyway and pay second-priceSomeone bids
105: you win; but you pay 105, i.e., 5 more than what you valueJan
07, 2009Game Theory2424
