Strategic Games - Applications

03. Strategic Games - Applications

Ryan Fang1

[email protected]

Decision Making and Strategy in EconomicsEcon 402 (Spring 2015)

Reading

Osborne: Chapter 3

1 Review the Key Concepts

2 Cournot Oligopolistic Competition

3 Bertrand Oligopolistic Competition

4 Hotellings Model of Electoral Competition

5 Auctions

A strategic Game

a set of players for each player, a set of actions

potentially different across players action profile: a list of actions, exactly one action from eachplayer.

for each player, a payoff function defined over the set of allpossible action profiles

Nash Equilibrium

A steady state action profile no player has an alternative action that can yield her astrictly higher payoff, given the actions of her opponents.

thus, no player has an incentive to deviate from a Nashequilibrium.

An action profile in which each players action is her bestresponse to the actions of her opponents.

if every player is already best responding, no one canprofitably deviate.

Dominance

An action of a player strictly dominates another action ofthe same player if, regardless of her opponents choices ofactions, the dominant action always yields her a strictlyhigher payoff than the dominated action.

An action of a player weakly dominates another action ofthe same player if:

i) regardless of her opponents choices of actions, thedominant action always yields her a payoff that is no lessthan that yielded by the dominated action; and,

ii) given some action profiles of her opponents, the dominantaction is strictly better than the dominant action.

Dominance and Nash Equilibria

A strictly dominated action of a player is never a bestresponse to any profiles of her opponents actions.

Thus, a strictly dominated action can never be played in anyNash equilibrium.

When looking for Nash equilibria, we can simply eliminatethe strictly dominated actions of every player.





5 Auctions

The Cournot Model

Two firms produce a homogeneous good. The cost to firm i = 1, 2 of producing qi units of the good isCi (qi) = c qi .

The output by the two firms are sold at a single price,determined by the inverse demand function. Specifically, wehave:

P (Q) =

{Q if Q 0 if Q >

where Q = q1 + q2 is the total output of the firms. Assumethat > c .

The Cournot Model

The profit of firm 1 given its choice of q1 and firm 2s choiceq2 is thus:

pi1 (q1, q2) = q1 (P (q1 + q2) c)

Similarly, the profit of firm 2 is:

pi2 (q1, q2) = q2 (P (q1 + q2) c)

The Cournot Game

We model the competition between the duopolists by thefollowing strategic game:

players: firm 1 and firm 2 Actions: each firm chooses a non-negative quantity Payoffs: each firms payoff is given by its profit given theoutputs of both firms

Best Responses

We would like to find the Nash equilibria of this game. To doso, we first solve for the firms best response functions.

For firm 1, we solve the following problem:

maxq1

q1 (P (q1 + q2) c)

For every q2 c , firm 1s profit is maximized by settingq1 = 12 ( c q2).

For q2 > c , firm 1s profit maximizing output is zero.

Best Responses

Summarizing, we have:

B1 (q2) =

{12 ( c q2) if q2 c0 if q2 > c

Similarly, firm 2s best response function is given by:

B2 (q1) =

{12 ( c q1) if q1 c0 if q1 > c

Nash Equilibrium

A Nash equilibrium of this game is a pair (q1, q2) satisfyingthe set of equations:{

q1 =12 ( c q2)

q2 =12 ( c q1)

The unique pair of (q1, q2) satisfying the two equations iswhere q1 = q2 =

13 ( c).

Comparison with Collusive Outcome

How does the total output in the Nash equilibrium comparewith the output produced by the two firms if they were tocollude and maximize the sum of their profits?

The sum of their profits is maximized by setting their totaloutput equal to the monopoly output.

What is the monopoly output level? Simply go back to eitherfirms best response function, and set their opponentsoutput level to zero.

B1 (0) =12( c 0)

Comparison with Collusive Outcome

Thus, the monopoly output level is 12 ( c). The totaloutput in the Nash equilibrium is 23 ( c), which is higher.

The firms can increase their profits if they were to agree onthe following collusive output levels:

(14 ( c) , 14 ( c)

).

Do you expect this to happen? Why?

n firms

Now suppose there are n firms, each with the same costfunction Ci (qi) = cqi .

The inverse demand function remains P (Q) = Q. The n firms simultaneously choose their output levels. What are the Nash equilibria of the game?

n firms

As before, we can solve for each firms best responsefunction.

For example, in the case of firm 1, we solve:

maxq1

q1 (P (q1 + q2 + . . .+ qn) c)

and find the unique maximizer for any (q2, q3, . . . , qn) to be:

B1 (q1) ={12 ( c

ni=2 qi) if

ni=2 qi c

0 ifn

i=2 qi > c

Nash Equilibrium

A Nash equilibrium of the game is a profile (q1, q2, . . . , qn)that satisfies:

q1 =12

( c i 6=1 qi )

q2 =12

( c i 6=1 qi )

qn =

12

( c i 6=n qi )

The unique Nash equilibrium of this game is whereq1 = q2 = . . . = qn = ( c) / (n + 1).

Increasing n

The total output of the n firms is nn+1 ( c), and themarket price is nn+1 ( c).

As n increases, the total output increases and the pricedecreases.

As n approaches , the total output approaches ( c)and the market price approaches c .

Thus, as n increases, the Nash equilibrium price and supplyapproach those in a perfectly competitive market.





5 Auctions

The Bertrand Model

In the Cournot model, firms choose their output levels.What if, instead of quantity competition, we have pricecompetition?

Suppose that the n = 2 firms each choose a price and areprepared to supply any quantity demanded from them attheir nominated prices.

Assume that the firms cost functions remain the same, thatis, Ci (qi) = cqi for i = 1, 2.

Assume also that the demand function is given by:

D (p) =

{ p if p 0 if p >

.

The Bertrand Model

The consumers always choose to purchase from the firm thatoffers the lowest price.

If the two firms offer the same price, then they each servehalf of the total demand for their goods.

Thus, firm i s profit is given by:

pii (p1, p2) =

(pi c) ( pi) if pi < pj12 (pi c) ( pi) if pi = pj0 if pi > pj

where j is i s competitor.

The Bertrand Game

We now model the competition between the duopolists bythe following strategic game:

Players: firm 1 and firm 2. Actions: each firm can choose any non-negative price (i.e.,Ai = R+).

Payoff: each players payoff is its profit given its own priceand that of its competitors.

What are the Nash equilibria of this game?

Finding Nash Equilibria

Claim: the unique Nash equilibrium of this game is wherep1 = p2 = c .

Proof: First we need to show that (c , c) is indeed a Nash equilibrium. Given that its opponent chooses price c , a firm can receive aprofit equal to 0.

Deviating to any price higher than c will yield a firm zeroprofits, while deviating to any price lower than c will yieldnegative profits.

Thus, there are no profitable deviations and (c , c) is a NE.


Proof (continued): There cannot be any Nash equilibria where at least one firmchooses a price lower than c . (why?)

There cannot be any Nash equilibria where at least one firmchooses a price higher than c . (why?)

What if the two firms prices are different? What if the two firms prices are equal?

Different Marginal Costs

Now suppose that firm 1s cost function is C1 (q1) = c1q1,while that of firm 2 is C2 (q2) = c2q2.

Without loss of generality, let c1 < c2. What are the Nash equilibria of the Bertrand game with thismodification?

Cournot or Bertrand?

The Nash equilibrium of the Cournot game and the Nashequilibrium of the Bertrand game are very different.

Which do you think is the better model? Why?





5 Auctions

Electoral Competition

What determines the number of candidates competing inelections?

What determines the candidates policy platforms? Why are there often no substantive differences between thepositions of political parties? (Not necessarily an accuratedescription of todays political landscape in the U.S.)

The Model

Assume that the policy space one-dimensional. Specifically,let it be the real line, R.

left wing vs. right wing There are two political parties, A and B, each choose apolicy position, xA, xB R.

There are a continuum of voters, each has an ideal policyposition, xi R.

The ideal positions of the voters are continuously distributedon R, and has a unique median point xm.

That is, exactly half the population has ideal points xm andhalf the population has ideal points xm.

The Model

We assume sincere voting. That is, the voters always votefor their most preferred policy platform.

Each voters preference over the policy positions, y R, isgiven by |y xi |.

Thus, a voter always votes for the party whose policyposition is closest to her own ideal point.

If both parties positions are equally distant from a votersideal position, she votes for either of them with equalprobability.

The Model

The two parties simultaneously choose their policy position.They care only about winning the election (i.e., they areoffice-seeking). So their payoff is assumed to be theprobability of winning.

The election is plurality rule: the party with the most voteswin.

Thus, a partys payoff is 1 (0) if strictly more (less) votersprefer their platform than that of their opponents. If equalnumber of voters strictly prefer either party, then each winsthe election with probability 12 .

What are the Nash equilibria of this game?


Claim: the unique Nash equilibrium of this game is whereboth parties choose the median voters ideal position, xm.

Proof: First, we check that (xm, xm) is a Nash equilibrium:

When both party choose xm, they split the votes in half, andeach obtains a payoff equal to 12 .

Given that its opponent chooses xm, any party deviating fromxm to some x < xm will only get the votes of those voterswhose ideal points fall in (, 12 (x + xm)] and hence lose theelection.

Similarly, deviating from xm to some x > xm will also lose aparty the election.

Therefore, (xm, xm) is a Nash equilibrium.


Proof (continued): Now consider any strategy profile where exactly one partychooses xm.

Then that party will win the election with certainty. Theother party can increase their payoff from 0 to 12 by deviatingto xm.


Proof (continued): Now suppose neither party chooses xm. Then either theirpositions are equally distant from xm or not.

In the first case, either party can increase their payoff from 12to 1 by moving slightly closer to xm.

In the second case, the party whose position is further fromxm can increase its payoff from 0 to 1 by deviating to aposition that is slightly closer to xm than its opponentsposition.

Therefore, there are no other Nash equilibria.

Discussion

Thus, the logic of political competition forces both party toadopt the same position, xm. Only then is neither party ableto increase its chances of winning.

Another way to look at this is that by adopting the positionxm, each party can guarantee that it does not lose theelection. That is, the parties are "maxminimizing."

If the opponent does the same thing, the "best" a player canhope for is also just "not losing".

This is what we call a zero-sum game. And in such games,the players always play maxminimizing strategies in any Nashequilibrium.





5 Auctions

Auctions

An auction is a mechanism for selling goods. Most commonly, potential buyers submit bids and the goodsare sold to the highest bidder.

However, the details of the rules involved can vary fromauctions to auctions. Different rules often result in verydifferent consequences in terms of revenue raised andallocation efficiency.

We are going to look at two types of auctions: second price(sealed bid) auctions and first price (sealed bid) auctions.

Second Price Auctions

In a second price (sealed bid) auction, also known as aVickrey auction, prospective buyers each submit a bid for asingle good.

The highest bidder obtains the good and pays the highestamong her opponents bids (hence the term "second price").

The Game

Let there be n 2 players. Player i values the good at vi . Assume that all players valuations are different and areordered such that v1 > v2 > . . . > vn.

Each player i chooses a bid, bi R+ (i.e., any non-negativereal number).

The payoffs to player i is vi max{bj}j 6=i if her bid is amongthe highest and she obtains the good and 0 otherwise.

When there are multiple bidders offering the highest bid, thegood is given to the player among these bidders who has thehighest valuation.

Nash Equilibria

There are many Nash equilibria of this game. First, consider the action profile where bi = vi for all i . This is a Nash equilibrium. Why? In this equilibrium, player 1 gets the good and pays v2.

Nash Equilibria

Another equilibrium is where b1 = v1 and bi = 0 for all i 6= 1.Why?

Here player 1 also gets the good, but pays 0.

Nash Equilibria

There are also Nash equilibria in which player 1 (who valuesthe good most) doesnt get the good.

Can you think of one?

Nash Equilibria


Can you think of one? What about (0, v1, 0, . . . , 0)?

Nash Equilibria


Can you think of one? What about (0, v1, 0, . . . , 0)? And (0, 0, . . . , 0, v1)?

Nash Equilibria


Can you think of one? What about (0, v1, 0, . . . , 0)? And (0, 0, . . . , 0, v1)? Do you find such Nash equilibria reasonable?

Nash Equilibria

The issue here is that Nash equilibrium permits weaklydominated actions.

recall that strictly dominated actions can never be a part ofany Nash equilibrium.

For example, in (0, v1, 0, . . . , 0), some players are playingweakly dominated strategies. Who?

Nash Equilibria

The issue here is that Nash equilibrium permits players playweakly dominated actions.

recall that strictly dominated actions can never be a part ofany Nash equilibrium.

For example, in (0, v1, 0, . . . , 0), some players are playingweakly dominated strategies. Who?

Everyone! Why?

Nash Equilibria

The unique weakly undominated action for player i is bi = vi . Any bi lower than vi is (weakly) dominated, since there areaction profiles of i s opponents where the highest bidamongst them is strictly lower than vi but strictly higherthan bi .

Any bi higher than vi is dominated, since there are actionprofiles of i s opponents where the highest bed amongstthem is strictly higher than vi but strictly lower than bi .

bi = vi for all i is the unique Nash equilibrium where allplayers play weakly undominated actions.

First Price Auctions

In a first price (sealed bid) auction, the players each submit abid.

The winner is still the one with the highest bid. However, the winner now has to pay her own bid. When there are ties, the good goes to the player amongst thehighest bidders who has the highest valuation for the good.

Nash Equilibria

There are again many Nash equilibria of this game. For example, (v2, v2, v3, . . . , vn) is a Nash equilibrium. Why?

Nash Equilibria

Claim: In any Nash equilibria, the good goes to player 1(who, recall, has the highest valuation of the good).

Proof:

Nash Equilibria

Claim: In any Nash equilibria, the good goes to player 1(who, recall, has the highest valuation of the good).

Proof: Suppose there is an equilibrium in which the winner issome other bidder i 6= 1.

It must be the case that the winners bid is no higher than v2.

But then player 1 can deviate to bidding bi and get a payoffv1 bi , which is strictly higher than 0, the payoff she getsshould she lose the bid.

Nash Equilibria

Claim: In any Nash equilibrium, the highest two bids must beequal and must be between v1 and v2.

Proof:

Nash Equilibria

Claim: In any Nash equilibrium, the two highest bids must beequal and must be between v1 and v2.

Proof: If the two highest bids are not equal, then the winning playercan strictly increase her payoff by lowering her bid slightly.

We already proved that player 1 must be the winner. Soplayer 1 must be one of the highest bidders. But player onecannot bid anything higher than v1 in equilibrium, for hewould be receiving a negative payoff.

Also, the highest bid cannot be lower than v2, because, if itwere the case, player 2 can bid slightly higher than thehighest bids and receive a positive payoff.

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Strategic Games - Applications