Game Theory

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Game Theory
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What is game theory (GT)?

• GT is the study of multi-agent decision problems
• GT is used to predict what will happen
  (equilibrium) when
• There are more than one agent but not too many
  for each of them to be negligible
• Each agents payoff depend on what they decide to
  do and what others decide to do
• Examples members of a team, oligopolies,
  international negotiations,

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Basic Textbook

• Game Theory for Applied Economists by Robert
  Gibbons
• The slides are also based in this textbook

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Elements of a game

• Players
• agents that take actions
• nature is also a player
• Information sets What is known by each agent at
  each moment that a decision must be taken
• Actions
• In each moment of a game, what can each agent
  choose. Examples q20,30,40 High, medium or low
  taxes
• Outcomes what is the result for each set of
  possible actions
• Payoffs Depending on the outcome, what each
  agent gets( could be in terms of utility or
  directly money)

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Static games of complete information
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What is a static game of complete information?

• Static (or simulateneous-move) game
• Players choose actions
• Each player chooses an action without knowing
  what the other players have choosen
• Complete information
• Every player will know the payoff that each agent
  will obtain depending on what actions have been
  taken

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The normal-form representation of a game

• Basic tool to analyze games
• Very useful for static games under complete
  information.
• It consists of
• The player in the game
• The strategies available to each player
• In this type of games, strategies are actions,
  but they will be very different in more
  complicated dynamic games !!!
• The payoff received by each player for each
  combination of strategies the could be chosen by
  the players
• Usually represented by a table

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The normal-form representation of a game
The Prisioners Dilemma Two prisoners. They are
being question by the police in different rooms.
Each can confess or not
Prisoner B
Prisoner A
Prisoner A
Be sure that you recognize the elements of the
game !!!!
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Solutions concepts for static games with complete
information

• Given a game, we will apply a solution concept to
  predict which will be the outcome (equilibrium)
  that will prevail in the game
• Elimination of strictly dominated strategies
• Nash equilibrium
• These solution concepts can also be applied to
  more complicated games but they are terribly
  useful for static games with complete information

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Elimination of strictly dominated strategies

• Intuitive solution concept
• Based on the idea that a player will never play a
  strategy that is strictly dominated by another
  strategy
• An strategy si of player i is strictly dominated
  by si if player is payoff is larger for si
  than for si independently of what the other
  players play!
• In the prisoners' dilemma, the strategy not
  confess is strictly dominated by confess

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Iterative Elimination of strictly dominated
strategies

• In some games, the iterative process of
  eliminating strictly dominated strategies lead us
  to a unique prediction about the result of the
  game (solution)
• In this case, we say that the game is solvable by
  iterative elimination of strictly dominated
  strategies
• Lets see an example

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Iterative Elimination of strictly dominated
strategies
If the red player is rational, it will never play
Right because it is strictly dominated by
Middle. If the blue player knows that red is
rational, then he will play as if the game were
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Iterative Elimination of strictly dominated
strategies
In this case if the blue player is rational and
he knows that red is rational, then the blue
player will never play Down. So, the red player
will play as if the game were
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Iterative Elimination of strictly dominated
strategies
So, if the red player is rational, and he know
that blue is rational, and he knows that blue
knows that he is rational, then the red player
will play Middle The solution of the game is
(UP, Middle)
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Problems with this solution concept
We need to assume that rationality is common
knowledge Decisions are tough. In many games
there are no strategies that are strictly
dominated or there are just a few and the
process of deletion does not take us to a
solution but only to a smaller game
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Example of a game were there are no dominated
strategies
In this game, no strategies are dominated, so the
concept of iterated elimination of dominated
strategies is not very useful Lets study the
other solution concept
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Some notation, before defining the solution
concept of Nash Equilibrium,

• SA strategies available for player A (a ? SA)
• SB strategies available for player B (b ? SB)
• UA utility obtained by player A when particular
  strategies are chosen
• UB utility obtained by player B when particular
  strategies are chosen

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Nash Equilibrium

• In games, a pair of strategies (a,b) is defined
  to be a Nash equilibrium if a is player As best
  strategy when player B plays b, and b is player
  Bs best strategy when player A plays a
• It has some resemblance with the market
  equilibrium where each consumer and producer were
  taking optimal decisions

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Nash Equilibrium in Games

• A pair of strategies (a,b) is defined to be a
  Nash equilibrium if
• UA(a,b) ? UA(a,b) for all a?SA
• UB(a,b) ? UB(a,b) for all b?SB

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Intuition behind Nash Eq.

• If a fortune teller told each player of a game
  that a certain (a,b) would be the predicted
  outcome of a game
• The minimum criterion that this predicted outcome
  would have to verify is that the prediction is
  such that each player is doing their best
  response to the predicted strategies of the other
  players that is the NE
• Otherwise, the prediction would not be internally
  consistent, would be unstable

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Intuition behind Nash Eq.

• If a prediction was not a NE, it would mean that
  at least one individual will have an incentive to
  deviate from the prediction
• The idea of convention if a convention is to
  develop about how to play a given game, then the
  strategies prescribed by the convention must be a
  NE, else at least one player will not abide by
  the convention

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Checking whether or not a pair of strategies is a
NE

• In the Prisioners Dilema
• (No Confess, No Confess)
• Notice that it is not optimal from the
  society-of-prisoners point of view
• In the previous 3x3 game BxR
• Notice that these NE also survived the iterated
  process of elimination of dominated strategies
  This is a general result

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Relation between NE and iterated

• If the process of iterative deletion of dominates
  strategies lead to a single solution this
  solution is a NE
• The strategies that are part of a NE will survive
  the iterated elimination of strictly dominated
  strategies
• The strategies that survive the iterated
  elimination of strictly dominated strategies are
  NOT necessarily part of a NE

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Finding out the NE of a game

• The underlining trick lets see it with previous
  games
• One cannot use this if the strategies are
  continuous (ie. Production level). We will see
  afterwards

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Multiple Nash Equilibrium
Some games can have more than one NE In this
case, the concept of NE is not so useful because
it does not give a clear predictionas in this
game called The Battle of the Sexes
SEX B
SEX A
Prisoner A
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Nash Equilibrium with continuous strategies
Example Duopoly. Firm i Firm j Strategies
Output level, that is continuous
Prisoner A
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Prisoner A
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One can draw the best response functions. The
intersection point is the NE
Prisoner A
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qj
a-c
Ri(qj)
(a-c)/2
(a-c)/3
Rj(qi)
a-c
qi
Prisoner A
(a-c)/2
(a-c)/3
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Nash Equilibrium in Mixed Strategies

• So far, we have used the word strategy. To be
  more explicit, we were referring to pure
  strategies
• We will also use the concept of mixed strategy
• In a static game with complete information, a
  mixed strategy is a vector that tell us with what
  probability the player will play each action that
  is available to him

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Nash Equilibrium in Mixed Strategies
Consider the following game matching pennies
Player 2
Player 1
Prisoner A
An example of a mixed strategy for player 1 would
be (1/3,2/3) meaning that player 1 will play
Heads with probability 1/3 and Tails with
probability 2/3 We can obviously say that a mixed
strategy is (q,1-q) where q is the probability of
Heads
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Nash Equilibrium in Mixed Strategies

• Why are Mixed Strategies useful?
• Because in certain games, players might find
  optimal to have a random component in their
  behavior
• For instance, if it was the case that the Inland
  Revenue would never inspect individuals taller
  than 190 cms, these individuals will have lots of
  incentives not to declare their income truthfully!

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Nash Equilibrium in Mixed Strategies
Notice that the matching pennies game does not
have an equilibrium in pure strategies
Player 2
Player 1
Prisoner A
Does this game have an equilibrium in mixed
strategies?
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Player 2
Player 1
Prisoner A
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Prisoner A
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Prisoner A
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Drawing the best responses
Notice the vertical and horizontal lines are
because of the any between 0,1
r
r(q) player 1 br
1
1
q(r) player 2 br
1/2
The Nash eq. in mixed strategies is the
intersection of the best responsein this
case (1/2,1/2) for player 1 and (1/2,1/2) for
player 2
0
1
1/2
q
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Existence of Nash Equilibrium

• A game could have more than one Nash Equilibrium
• The same game could have equilibria in both pure
  and mixed strategies or only pure or only mixed
• Notice that this is a bit artificial any pure
  strategy is a mixed strategy where one action has
  probability 1
• Any game has at least one NE, but this one could
  be in mixed strategies

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Dynamic games of complete information
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Extensive-form representation -In dynamic games,
the normal form representation is not that
useful. The extensive-form representation will be
a very useful tool in this setting. It consists
of -players -when each player has the
move -what each player can do at each of his
opportunities to move -what each player knows
at each of his opportunities to move -the
payoff received by each player for each
combination of moves that could be chosen by the
players -Usually represented by a tree
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Example of Extensive form representation
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Strategies for dynamic games -In dynamic games,
we have to be much more careful defining
strategies. A strategy is a complete plan of
action it specifies a feasible action for the
player in every contingency in which the player
might be called on to act In the previous
example, a strategy for player A is (L). Another
possible strategy for player A is (H) An example
of a strategy for player B is (L,S) that means
that player B will play L if he gets to his first
node and will play S if he get to the second
node. Other strategies would be (L,L) (S,S) and
(S,L)
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Extensive-form representation -It could be that
when a player moves, he cannot distinguish
between several nodes he does not know in what
node he is! -For instance, it could be that when
player B moves, he has not heard the noise from
player A -We reflect this ignorance by putting
these two nodes together in the same circle as
in the following slide
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A Dormitory Game
Notice that this game will actually be static
!!!!!!
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• Information set
• -This take us to the notion of an information
  set!!
• An information set for a player is a collection
  of decision nodes satisfying
• The player has the move at every node in the
  information set
• When the play of the game reaches a node in the
  information set, the player with the move does
  not know which node in the information set has
  (or has not) been reached
• As a consequence, the nodes surrounded by the
  same circle are part of the same information set

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Strategies -We can be more precise defining what
a strategy is. A players strategy is an action
for each information set that the player has !!!
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Dynamic games with complete information Can be
divided in Perfect information at each move in
the game, the player with the move knows the full
history of the play of the game so far Imperfect
information at some move the player with the
move does not know the history of the
game Notice, in complete games with perfect
information each information set must have one
and only one node (the information set is
singleton). If info is complete but there is an
information set with more than one node, it must
be an imperfect information game.
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Dynamic games with complete information Another
classification Non-repeated games The game is
just played once Finitely repeated games The
game is repeated a finite number of
times Infinitely repeated games The game is
repeated an infinite amount of times
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Non-repeated dynamic games with perfect
information
Two main issues -Is the Nash Equilibrium an
appropriate solution concept? -If not Define a
better solution concept
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A Two-Period Dormitory Game
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A Two-Period Dormitory Game

• Each strategy is stated as a pair of actions
  showing what B will do depending on As actions

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A Two-Period Dormitory Game

• There are 3 Nash equilibria in this game
• AL, B(L,L)
• AL, B(L,S)
• AS, B(S,L)

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A Two-Period Dormitory Game

• B (a,b)
• a strategy that B plays if A plays L
• b strategy that B plays if A plays S
• AL, B(L,S) and AS, B(S,L) do not seem
  appropiate
• each incorporates a non-credible threat on the
  part of B (out of the equilibrium path)
• For instance regarding AL, B(L,S), If A chose
  S out of equilibrium- it is not credible that B
  chose S as (L,S) indicates

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• In games with more than one period, there might
  be strategies that are Nash Eq but they involve
  no credible threats
• We need a concept of equilibrium for games with
  more than one period
• The concept will be called Subgame Perfect Nash
  Equilibrium (SPNE)

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• We will define SPNE more formally later on. For
  the time being, lets say that
• A SPNE is a Nash equilibrium in which the
  strategy choices of each player do not involve
  no-credible threats
• A strategy involves no-credible threats if they
  require a player to carry out an action that
  would not be in its interest at the time the
  choice must be made

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• A simple way to obtain the SPNE in dynamic games
  with perfect information is to solve the game
  backwards, called backwards induction
• Whiteboard with the dormitory example
• Algorithm
• Find the optimal action at each of the
  predecessors of the terminal nodes
• Associate these nodes with the payoffs of the
  anticipated terminal node
• Start again the process with this reduced game
• (see another description of the algorithm in the
  distributed handout)

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• Example in distributed handout
• Another example, Pg. 60 of the book
• The (c,s) example
• Of the three NE that we had in the Dormitory
  game, only B(L,L), A L is a SPNE
• In the backward induction procedure, we are
  asking each individual to do whatever is best
  whenever they move (independently whether they
  are or not in the equilibrium path) so it is not
  possible to have non-credible threats

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• Backwards Induction with continuous strategies
• Example Stackelberg model of duopoly
• Firm 1 produces q1
• Firm 2, observes q1 and produces q2
• Compute R(q1) Firm 2s optimal response to an
  arbitrary level of production by Firm 1.
• R(q1) is Firm 2s best response.
• Compute what is the optimal q1 for Firm 1 if she
  know that Firm 2 will produce R(q1)
• Pa-b(q1R(q1))

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• Backwards Induction with continuous strategies
• Example Stackelberg model of duopoly
• Firm 1 produces q1
• Firm 2, observes q1 and produces q2
• Compute R(q1) Firm 2s optimal response to an
  arbitrary level of production by Firm 1.
• R(q1) is Firm 2s best response.
• Compute what is the optimal q1 for Firm 1 if she
  know that Firm 2 will produce R(q1)
• Pa-b(q1R(q1))

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• Non-repeated dynamic games complete but imperfect
  information
• At some point of the game, a player does not know
  exactly in which node he or she is (does not
  completely know the history of moves)
• See example
• We cannot apply Backwards Induction because a
  player might not know what is best to play as she
  might not know in what node she is
• In order to understand the solution concept, we
  must define a SPNE more formally.
• To do that, we must understand what a subgame is

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• Non-repeated dynamic games complete but imperfect
  information
• A subgame in an extensive form game
• Begins at a decision node n that is a singleton
  (but is not the games first decision node)
• Includes all the decision and terminal nodes
  following n in the game tree (but no nodes that
  do not follow n)
• Does not cut any information sets, that is, if a
  decision node n follows n in the game tree, then
  all other nodes in the information set containing
  n must also follow n, and so must be included in
  the subgame
• See example in paper
• See pg 121 in the book

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• Non-repeated dynamic games complete but imperfect
  information
• A strategy profile is SPNE
• If it is a Nash Equilibrium of the game and of
  every subgame of the game
• Two ways to find the SPNE of a game
• Obtain the NE and then see which of them imply
  that that they are NE of the different subgames
  of the game
• Finding the NE of the last information sets and
  substitute backwards
• See example in the handout

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Repeated Games

• Many economic situations can be modeled as games
  that are played repeatedly
• consumers regular purchases from a particular
  retailer
• firms day-to-day competition for customers
• workers attempts to outwit their supervisors

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Repeated Games

• An important aspect of a repeated game is the
  expanded strategy sets that become available to
  the players
• opens the way for credible threats and subgame
  perfection
• It is important whehter or not the game is
  repeated a finite or infinite number of times

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Prisoners Dilemma Finite Game

• Firms A and B. Low or High price. In a one shot
  game, (L,L) no cooperating- is the NE

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Prisoners Dilemma Finite Game

• The NE is inferior to (H,H) the cooperating
  strategy

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Prisoners Dilemma Finite Game

• Suppose this game is to be repeatedly played for
  a finite number of periods (T)
• Any expanded strategy in which A promises to play
  H in the final period is not credible
• when T arrives, A will choose strategy L
• The same logic applies to player B

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Prisoners Dilemma Finite Game

• Any SPNE for this game can only consist of the
  Nash equilibrium strategies in the final round
• AL BL
• The logic that applies to period T also applies
  to period T-1
• Do backward induction in the whiteboard
• The only SPNE in this finite game is to require
  the Nash equilibrium in every round -gt No
  cooperation

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Eq. in a Finite Repeated Game

• If the one-shot game that is repeated a finite
  number of times has a unique NE then the game
  repeated game has a unique outcome the NE of the
  one-shot game

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Game with Infinite Repetitions

• We cannot use backward induction because there is
  no a terminal node
• In Infinite games, each player can announce a
  trigger strategy
• promise to play the cooperative strategy as long
  as the other player does
• when one player deviates from the pattern, the
  other player will play no cooperation in the
  subsequent periods and hence the game will
  revert to the single period NE

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Game with Infinite Repetitions

• Lets think of a players decision in any
  arbitrary node of the game
• If B decides to play cooperatively, payoffs of 2
  can be expected to continue indefinitely
• If B decides to cheat, the payoff in period K
  will be 3, but will fall to 1 in all future
  periods

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• Lets think of a players decision in any
  arbitrary node of the game
• If ? is player Bs discount rate, the present
  value of continued cooperation is
• 2 ?2 ?22 2/(1-?)
• The payoff from cheating is
• 3 ?1 ?21 3 1/(1-?)
• Continued cooperation will be credible (will be a
  NE of the subgame that starts in the node where
  the player is choosing) if
• 2/(1-?) gt 3 1/(1-?)
• gt ½

• If players value the future enough, they will
  prefer to cooperate in the case of firms this is
  called Tacit Collusion

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Game with Infinite Repetitions

• ? can also be interpreted as the probability that
  the game will continue one more period
• (Folk theorem) Let G be a finite, static game of
  complete information. Let (e1,e2,,en) denote the
  payoffs from a NE of G, and let (x1,x2,,xn)
  denote any other feasible payoffs from G. If
  xigtei foor every player i and if ? is
  sufficiently close to one, then there exists a
  SPNE of the infinitely repeated game that
  achieves (x1,x2,xn) as the average payoff
• In other words, if cooperation is better than the
  NE for every player and players value the future
  enough then cooperation is a SPNE of the game
  repeated a infinite number of times

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Games of Incomplete Information

• There is at least one player that does not know
  the payoff of at least one player

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Games of Incomplete Information

• Each player in a game may be one of a number of
  possible types (tA and tB)
• player types can vary along several dimensions
• We will assume that our player types have
  differing potential payoff functions
• each player knows his own payoff but does not
  know his opponents payoff with certainty

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Games of Incomplete Information

• Each players conjectures about the opponents
  player type are represented by belief functions
  fA(tB)
• consist of the players probability estimates of
  the likelihood that his opponent is of various
  types
• Games of incomplete information are sometimes
  referred to as Bayesian games

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Games of Incomplete Information

• The payoffs to A and B depend on the strategies
  chosen (a ? SA, b ? SB) and the player types
• For one-period games, it is fairly easy to
  generalize the Nash equilibrium concept to
  reflect incomplete information
• we must use expected utility because each
  players payoffs depend on the unknown player
  type of the opponent

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Games of Incomplete Information

• A strategy pair (a,b) will be a Bayesian-Nash
  equilibrium if a maximizes As expected utility
  when B plays b and vice versa

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A Bayesian-Cournot Equilibrium

• Suppose duopolists compete in a market for which
  demand is given by
• P 100 qA qB
• Suppose that MCA MCB 10
• the Nash (Cournot) equilibrium is qA qB 30
  and payoffs are ?A ?B 900

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A Bayesian-Cournot Equilibrium

• Suppose that MCA 10, but MCB may be either high
  ( 16) or low ( 4)
• Suppose that A assigns equal probabilities to
  these two types for B so that the expected MCB
  10
• B does not have to consider expectations because
  it knows there is only a single A type

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A Bayesian-Cournot Equilibrium

• B chooses qB to maximize
• ?B (P MCB)(qB) (100 MCB qA qB)(qB)
• The first-order condition for a maximum is
• qB (100 MCB qA)/2
• Depending on MCB, this is either
• qBH (84 qA)/2 or
• qBL (96 qA)/2

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A Bayesian-Cournot Equilibrium

• Firm A must take into account that B could face
  either high or low marginal costs so its expected
  profit is
• ?A 0.5(100 MCA qA qBH)(qA)
  0.5(100 MCA qA qBL)(qA)
• ?A (90 qA 0.5qBH 0.5qBL)(qA)

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A Bayesian-Cournot Equilibrium

• The first-order condition for a maximum is
• qA (90 0.5qBH 0.5qBL)/2
• The Bayesian-Nash equilibrium is
• qA 30
• qBH 27
• qBL 33
• These choices represent an ex ante equilibrium

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Dynamic Games with Incomplete Information

• In multiperiod and repeated games, it is
  necessary for players to update beliefs by
  incorporating new information provided by each
  round of play
• Each player is aware that his opponent will be
  doing such updating
• must take this into account when deciding on a
  strategy
• We will not study them in this course