Wireless Network Pricing Chapter 6: Oligopoly Pricing Jianwei Huang - - PowerPoint PPT Presentation

Wireless Network Pricing Chapter 6: Oligopoly Pricing

Jianwei Huang & Lin Gao

Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong

The Book

E-Book freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Physical book available for purchase from Morgan & Claypool (http://goo.gl/JFGlai) and Amazon (http://goo.gl/JQKaEq)

Chapter 6: Oligopoly Pricing

Focus of This Chapter

Key Focus: This chapter focuses on the user interactions in an

• ligopoly market, where multiple self-interested individuals make

decisions independently, and the payoff of each individual depends not only on his own decision, but also on the decisions of others. Theoretic Approach: Game Theory

Game Theory

Follow the discussions in

Definition (Game Theory) Game theory is a study of strategic decision making. Specifically, it is the study of mathematical models of conflict and cooperation between intelligent rational individuals.

Section 6.1 Theory: Game Theory

What is a game?

A game is a formal representation of a situation in which a number of individuals interact with strategic interdependence.

• n the choices of other individuals;

actions that produce his most preferred outcome.

Key components of game

decisions? What information do players know about each other when making decisions?

combinations chosen by players?

possible outcomes?

Strategic Form Game

In strategic form games (also called normal form games), all players make decisions simultaneously without knowing each other’s choices. Definition (Strategic Form Game) A strategic form game is a triplet hI, (Si)i2I, (ui)i2Ii where I = {1, 2, ..., I} is a finite set of players; Si is a set of available actions (pure strategies) for player i 2 I;

ui : S ! R is the payoff (utility) function of player i, which maps every possible action profile in S to a real number.

Prisoner's dilemma

• Two suspects are arrested.
• The police lack sufficient evidence to convict

the suspects, unless at least one confesses.

• The police hold the suspects in separate

rooms, and tell each of them three possible consequences.

Prisoner's dilemma

• If both deny: 1 month in jail each.
• If both confess: 6 months in jail each.
• If one confesses and one denies
• The one confesses: walk away free of charge.
• The one denies: serve 12 months in jail.

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Player 2

Deny Confess

Player 1

Deny Confess

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Lady

Deny Confess

Gentleman

Deny Confess

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Player 2

Deny Confess

Player 1

Deny Confess

Prisoner's dilemma

Player 2

Deny

Player 1

Deny Confess

• 1, -1

0, -12

Prisoner's dilemma

Player 2

Confess

Player 1

Deny Confess

• 12, 0
• 6, -6

Strictly Dominant

• Confess is a strictly dominant strategy for

player 1, as it always leads to the best payoff among all his strategies, independent of player 2’s strategy.

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Player 2

Deny Confess

Player 1

Deny Confess

Prisoner's dilemma

Player 2

Deny Confess

Player 1

Deny

• 1, -1
• 12, 0

Prisoner's dilemma

Player 2

Deny Confess

Player 1

Confess

0, -12

• 6, -6

Strictly Dominant

• Confess is also a strictly dominant strategy for

player 2.

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Player 2

Deny Confess (dominant)

Player 1

Deny Confess (dominant)

Prisoner's dilemma

Player 2

Deny Confess (dominant)

Player 1

Confess (dominant)

0, -12

• 6, -6

Prisoner's dilemma

Player 2

Confess (dominant)

Player 1

Confess (dominant)

• 6, -6

Prisoner's dilemma

• 1, -1
• 12, 0

0, -12

• 6, -6

Player 2

Deny Confess (dominant)

Player 1

Deny Confess (dominant)

Prisoner's dilemma

• In Prisoner’s dilemma, individual strategic
• ptimizations lead to a “lose-lose” situation.
• (deny, deny) leads to better payoffs than (confess,

confess).

• Such a result is stable: no player has incentive

to change the strategy.

• When there is a strictly dominant strategy, a

player can safely remove all other strategies.

Strictly Dominated

• Most games do not have strictly dominant

strategies.

• We can look for strictly dominated strategies.
• For player 1, strategy x is strictly dominated,

if choosing x always leads to a strictly less payoff than choosing another strategy y, independent of player 2’s choice.

Strategic Form Game

Definition (Strictly Dominated Strategy) For a strategy si 2 Si, if there exists some strategy s0

i 2 Si such that

ui(si, si) < ui(s0

i, si),

8si 2 Si, then strategy si is strictly dominated by strategy s0

i.

A strictly dominated strategy can be safely removed from the player’s strategy set without changing the game outcome.

An Example

1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 2

Left Middle

Player 1

Up Down Right

An Example

1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 2

Left Middle

Player 1

Up Down Right (dominated)

Strictly Dominated

• A strictly dominated strategy may not be the

worst

• As long as another strategy is strictly better than it.
• We can safely remove a strictly dominated

strategy, as it will never be played.

• This can be done iteratively until all strictly

dominated strategies are eliminated.

An Example

1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 2

Left Middle

Player 1

Up Down Right (dominated)

An Example

Player 2

Left Middle

Player 1

Up Down

1, 0 1, 2 0, 3 0, 1

An Example

Player 2

Left Middle

Player 1

Up Down (dominated)

1, 0 1, 2 0, 3 0, 1

An Example

Player 2

Left Middle

Player 1

Up

1, 0 1, 2

An Example

Player 2

Middle

Player 1

Up

1, 2

An Example

1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 Player 2

Left Middle

Player 1

Up Down Right

Iterative Elimination

• Any strictly dominated strategy can be

removed without affecting the game outcome.

• If there are multiple strictly dominated

strategies, they can be removed in any order.

• New strictly dominated strategy might

emerge during the removing process.

Finding Equilibrium

• When there are no strictly dominant or strictly

dominated strategies, we can not easily eliminate any strategies.

• We need to understand the best choices of

players considering each other’s choices.

• We try to find a stable outcome where both

players are happy, hence an equilibrium.

Strategic Form Game

Best Response Correspondence

• utcome for a player, taking all other players’ strategies as given.

Definition (Best Response Correspondence) For each player i, the best response correspondence Bi(si) : Si ! Si is a mapping from the set Si into Si such that Bi(si) = {si 2 Si | ui(si, si) ui(s0

i, si), 8s0 i 2 Si}.

Stag Hunt

• Two individuals go for a hunt.
• Each one can hunt a stag (deer) or a hare.
• Successful hunt of stag requires cooperation.
• Successful hunt of hare can be done

individually.

• Simultaneous decisions without prior

communications.

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Stag Hunt

• There is no strictly dominant or strictly

dominated strategies.

• We will find out a player’s best response

given the other player’s choice.

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Stag Hunt

Player 2

Stag

Player 1

Stag Hare

5, 5 2, 0

Stag Hunt

Player 2

Stag

Player 1

Stag Hare

5, 5 2, 0

Stag Hunt

Player 2

Hare

Player 1

Stag Hare

0, 2 2, 2

Stag Hunt

Player 2

Hare

Player 1

Stag Hare

0, 2 2, 2

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Nash Equilibrium (NE)

• A pair of strategies form a Nash Equilibrium

(NE) if each player is choosing the best response given the other player’s strategy choice.

• At a Nash equilibrium, no player can perform

a profitable deviation unilaterally.

Strategic Form Game

Nash Equilibrium

the incentive to change his strategy unilaterally.

Definition (Pure Strategy Nash Equilibrium) A pure strategy Nash Equilibrium of a strategic form game hI, (Si)i2I, (ui)i2Ii is a strategy profile s⇤ 2 S such that for each player i 2 I, the following condition holds ui(s⇤

i , s⇤ i) ui(s0 i, s⇤ i),

8s0

i 2 Si.

Equilibrium Selection

• How to choose between two Nash equilibria?
• (Stag, Stag) is payoff dominant: both players

get the best payoff possible.

• (Hare, Hare) is risk dominant: minimum risk

if player is uncertain of each other’s choice.

• Many theories, open problem.

Battle of Sexes

• A couple need to decide where to go during

Friday night.

• Husband prefers to go and watch football.
• Wife prefers to go and watch ballet.
• Both prefer to stay together during the night.
• They will make simultaneous decisions during

the day without prior communications.

Battle of Sexes

4, 2 0, 0 0, 0 2, 4 Wife

Football Ballet

Husband

Football Ballet

Battle of Sexes

Wife

Football

Husband

Football Ballet

4, 2 0, 0

Battle of Sexes

Wife

Football

Husband

Football Ballet

4, 2 0, 0

Battle of Sexes

Wife

Ballet

Husband

Football Ballet

0, 0 2, 4

Battle of Sexes

Wife

Ballet

Husband

Football Ballet

0, 0 2, 4

Battle of Sexes

4, 2 0, 0 0, 0 2, 4 Wife

Football Ballet

Husband

Football Ballet

Battle of Sexes

Wife

Football Ballet

Husband

Football

4, 2 0, 0

Battle of Sexes

Wife

Football Ballet

Husband

Football

4, 2 0, 0

Battle of Sexes

Wife

Football Ballet

Husband

Ballet

0, 0 2, 4

Battle of Sexes

Wife

Football Ballet

Husband

Ballet

0, 0 2, 4

Battle of Sexes

4, 2 0, 0 0, 0 2, 4 Wife

Football Ballet

Husband

Football Ballet

Anti-Coordination Game

• In Anti-Coordination Game, it is beneficial for

players to choose different strategies.

Hawk-Dove Game

• Two birds flight over a valuable territory.
• Two possible strategies:
• Hawk: flight until injured or your opponent retreats.
• Dove: display hostility, but retreat if your opponent

chooses to fight.

Hawk-Dove Game

• 2, -2

2, 0 0, 2 1, 1 Player 2

Hawk Dove

Player 1

Hawk Dove

Hawk-Dove Game

Player 2

Hawk

Player 1

Hawk Dove

• 2, -2

0, 2

Hawk-Dove Game

Player 2

Hawk

Player 1

Hawk Dove

• 2, -2

0, 2

Hawk-Dove Game

Player 2

Dove

Player 1

Hawk Dove

2, 0 1, 1

Hawk-Dove Game

Player 2

Dove

Player 1

Hawk Dove

2, 0 1, 1

Hawk-Dove Game

• 2, -2

2, 0 0, 2 1, 1 Player 2

Hawk Dove

Player 1

Hawk Dove

Hawk-Dove Game

• 2, -2

2, 0 0, 2 1, 1 Player 2

Hawk Dove

Player 1

Hawk Dove

Hawk-Dove Game

• 2, -2

2, 0 0, 2 1, 1 Player 2

Hawk Dove

Player 1

Hawk Dove

Pure Strategy NE

• So far we have restricted each player to choose
• ne strategy.
• Also called pure strategy.
• The corresponding Nash equilibrium is called

pure strategy Nash equilibrium (PNE).

Mixed Strategies

• Sometimes a game may not have a pure

strategy Nash equilibrium.

• We will allow players to “randomize” over

their strategies.

• We will see that it can be very natural to do so.

Matching Pennies

• Two individuals, each having a penny.
• They simultaneously decide to show head or

tail of their own penny.

• One player prefers to have a matching result.
• The other player prefers mismatch.

Matching Pennies

1, -1

• 1, 1
• 1, 1

1, -1 Player 2

Head Tail

Player 1

Head Tail

Matching Pennies

Player 2

Head Tail

Player 1

Head Tail

1, -1

• 1, 1

Matching Pennies

Player 2

Tail

Player 1

Head Tail

• 1, 1

1, -1

Matching Pennies

Player 2

Head Tail

Player 1

Head

1, -1

• 1, 1

Matching Pennies

Player 2

Head Tail

Player 1

Tail

• 1, 1

1, -1

Matching Pennies

1, -1

• 1, 1
• 1, 1

1, -1 Player 2

Head Tail

Player 1

Head Tail

Matching Pennies

• There is no pure strategy Nash equilibrium.
• Given an opponent’s pure strategy, a player

always has a profitable deviation:

• Player 1 Head -> Player 2 Tail -> Player 1 Tail ->

Player 2 Head -> Player 1 Head -> ...

• To prevent the opponent from taking

advantage, it is better to “randomize” the choices.

• Assume that player 2 plays

Head half of the time and Tail half of the time.

• Player 1’s expected payoff of

choosing Head is:

• 1*(1/2) + (-1) *(1/2) = 0
• Player 1’s expected payoff of

choosing Tail is:

• (-1)*(1/2) + 1 *(1/2) = 0

Matching Pennies

Player 2

Head (1/2) Tail (1/2)

Player 1

Head Tail

1, -1

• 1, 1
• 1, 1

1, -1

Matching Pennies

• Player 1 is indifferent between Head and Tail,

when player 2 chooses according to (1/2, 1/2).

• In fact, if player 1 chooses Head with a

probability p and Tail with a probability 1-p, then the expected payoff is p*0 + (1-p)*0 = 0.

• Hence choosing the two strategies based on (p,

1-p) for any p is player 1’s best response.

• Similarly, assume that player

1 plays Head half of the time and Tail half of the time.

• Player 2’s expected payoff of

choosing Head is:

• (-1)*(1/2) + 1 *(1/2) = 0
• Player 2’s expected payoff of

choosing Tail is:

• 1*(1/2) + (-1) *(1/2) = 0

Matching Pennies

Player 2

Head Tail

Player 1

Head (1/2) Tail (1/2)

1, -1

• 1, 1
• 1, 1

1, -1

Matching Pennies

• Player 2 is indifferent between Head and Tail,

when player 1 chooses according to (1/2, 1/2).

Matching Pennies

• (1/2, 1/2) and (1/2, 1/2) are mutual best

responses, since no player can find a profitable deviation.

• Unique Mixed strategy Nash equilibrium

(MNE): ((1/2, 1/2), (1/2, 1/2)).

Strategic Form Game

Mixed Strategy

mass function) over all pure strategies of a player.

1 can be σ1 = (0.4, 0.6), which means that player 1 picks “HEADS” with probability 0.4 and “TAILS” with probability 0.6.

ui(σ) = X

s2S

• ΠI

j=1σj(sj)

• · ui(s),

Strategic Form Game

Mixed Strategy Nash Equilibrium

under which no player has the incentive to change his mixed strategy unilaterally.

Definition (Mixed Strategy Nash Equilibrium) A mixed strategy profile σ⇤ is a mixed strategy Nash Equilibrium if for every player i 2 I, ui(σ⇤

i , σ⇤ i) ui(σ0 i, σ⇤ i),

8σ0

i 2 Σi.

Nash Equilibrium: σ⇤ = (σ⇤

1, σ⇤ 2) with σ⇤ i = (0.5, 0.5), i = 1, 2.

Strategic Form Game

“Support” of Mixed Strategy

which are assigned positive probabilities. That is, supp(σi) , {si 2 Si | σi(si) > 0}.

Theorem A mixed strategy profile σ⇤ is a mixed strategy Nash Equilibrium if and

• nly if for every player i 2 I, the following two conditions hold:

Every chosen action is equally good, that is, the expected payoff given σ⇤

i of every si 2 supp(σi) is the same;

Every non-chosen action is no better, that is, the expected payoff given σ⇤

i of every si /

2 supp(σi) must be no larger than the expected payoff of si 2 supp(σi).

How to Compute MNE

• In general, how do we compute MNE?
• In fact, the three games all have MNE.
• Stag Hunt
• Battle of Sexes
• Hawk-Dove
• Let’s compute two of them.

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag Hare

Player 1

Stag Hare

Stag Hunt

5, 5 0, 2 2, 0 2, 2 Player 2

Stag (p) Hare (1-p)

Player 1

Stag (q) Hare (1-q)

• Given p, Player 1 is indifferent

between Stag and Hare.

• Player 1’s expected payoff of

choosing Stag is 5*p + 0 *(1-p).

• Player 1’s expected payoff of

choosing Hare is 2*p + 2 *(1-p).

• 5*p + 0 *(1-p) = 2*p + 2 *(1-p)

Hence p = 0.4

• Due to symmetry, q = 0.4.

Stag Hunt

Player 2 Player 1

5, 5 0, 2 2, 0 2, 2

Stag (p) Hare (1-p) Stag Hare

Unique MNE

5, 5 0, 2 2, 0 2, 2 Player 2

Stag (0.4) Hare (0.6)

Player 1

Stag (0.4) Hare (0.6)

Battle of Sexes

4, 2 0, 0 0, 0 2, 4 Wife

Football Ballet

Husband

Football Ballet

Battle of Sexes

4, 2 0, 0 0, 0 2, 4 Wife

Football (p) Ballet (1-p)

Husband

Football (q) Ballet (1-q)

• Given p, husband is indifferent

between football and ballet.

• Husband’s expected payoff of

choosing football is 4*p + 0 *(1-p).

• Husband’s expected payoff of

choosing ballet is 0*p + 2 *(1-p).

• 4*p + 0 *(1-p) = 0*p + 2 *(1-p)

Hence p = 1/3

Battle of Sexes

Wife Husband

4, 2 0, 0 0, 0 2, 4

Football (p) Ballet (1-p) Football Ballet

• Given q, wife is indifferent

between football and ballet.

• Wife’s expected payoff of

choosing football is 2*q + 0 *(1-q).

• Wife’s expected payoff of

choosing ballet is 0*q + 4 *(1-q).

• 2*q + 0 *(1-q) = 0*q + 4 *(1-q)

Hence q = 2/3

Battle of Sexes

Wife Husband

4, 2 0, 0 0, 0 2, 4

Football Ballet Football (q) Ballet (1-q)

Battle of Sexes: Unique MNE

4, 2 0, 0 0, 0 2, 4 Wife

Football (1/3) Ballet (2/3)

Husband

Football (2/3) Ballet (1/3)

Computing MNE Payoff

4, 2 0, 0 0, 0 2, 4 Wife

Football (1/3) Ballet (2/3)

Husband

Football (2/3) Ballet (1/3)

2/9 1/9 4/9 2/9

Computing MNE Payoff

• Husband gets an expected payoff of

2/9*4 + 4/9*0 + 1/9*0 + 2/9*2 = 12/9 = 4/3

• Wife gets an expected payoff of

2/9*2 + 4/9*0 + 1/9*0 + 2/9*4 = 12/9 = 4/3

Comparing Payoffs

Strategy Profile Payoffs Pure Nash Equilibrium 1 (Football, Football) (4, 2) Pure Nash Equilibrium 2 (Ballet, Ballet) (2, 4) Mixed Nash Equilibrium ((2/3,1/3), (1/3,2/3)) (4/3, 4/3)

Strategic Form Game

Existence of Nash Equilibrium

strategy Nash equilibrium?

Theorem (Existence (Nash 1950)) Any finite strategic game, i.e., a game that has a finite number of players and each player has a finite number of action choices, has at least one mixed strategy Nash Equilibrium.

Strategic Form Game

Theorem (Existence (Debreu-Fan-Glicksburg 1952)) The strategic form game hI, (Si)i2I, (ui)i2Ii has a pure strategy Nash equilibrium, if for each player i 2 I the following condition hold: Si is a non-empty, convex, and compact subset of a finite-dimensional Euclidean space. ui(s) is continuous in s and quasi-concave in si. Compact: closed and bounded. Quasi-concave: a function f (·) is quasi-concave if f (·) is quasi-convex