UC BerkeleyDepartment of Economics

Game Theory in the Social Sciences(Econ C110)Fall 2016

Strategic games II

Oct 17, 2016

Some classical applications

— The Tragedy of the Commons.

— Oligopolistic competition.

— Sealed-bid auctions.

The tragedy of the commons

William Forster Lloyd (1833)

— Cattle herders sharing a common parcel of land (the commons) onwhich they are each entitled to let their cows graze. If a herder putmore than his allotted number of cattle on the common, overgrazingcould result.

— Each additional animal has a positive effect for its herder, but the costof the extra animal is shared by all other herders, causing a so-called“free-rider” problem. Today’s commons include fish stocks, rivers,oceans, and the atmosphere.

Garrett Hardin (1968)

— This social dilemma was populated by Hardin in his article “The Tragedyof the Commons,” published in the journal Science. The essay derivedits title from Lloyd (1833) on the over-grazing of common land.

— Hardin concluded that “...the commons, if justifiable at all, is justifi-able only under conditions of low-population density. As the humanpopulation has increased, the commons has had to be abandoned inone aspect after another.”

— “The only way we can preserve and nurture other and more preciousfreedoms is by relinquishing the freedom to breed, and that very soon.“Freedom is the recognition of necessity” — and it is the role of ed-ucation to reveal to all the necessity of abandoning the freedom tobreed. Only so, can we put an end to this aspect of the tragedy of thecommons.”

“Freedom to breed will bring ruin to all.”

Let’s put some game theoretic analysis (rigorous sense) behind this story:

— There are players, each choosing how much to produce in a produc-tion activity that ‘consumes’ some of the clean air that surrounds ourplanet.

— There is amount of clean air, and any consumption of clean aircomes out of this common resource. Each player = 1 chooseshis consumption of clean air for production ≥ 0 and the amount ofclean air left is therefore

−X

=1

— The benefit of consuming an amount ≥ 0 of clean air gives player a benefit equal to ln(). Each player also enjoys consuming thereminder of the clean air, giving each a benefit

ln³ −

X

=1´

— Hence, the value for each player from the action profile (outcome) = (1 ) is give by

( −) = ln() + lnµ −

X

=1

¶

— To get player ’s best-response function, we write down the first-ordercondition of his payoff function:

( −)

=1

− 1

−P=1

= 0

and thus

(−) = −P

6= 2

The two-player Tragedy of the Commons

— To find the Nash equilibrium, there are equations with unknownthat need to be solved. We first solve the equilibrium for two players.Letting () be the best response of player , we have two best-response functions:

1(2) = − 22

and 2(1) = − 12

— If we solve the two best-response functions simultaneously, we find theunique (pure-strategy) Nash equilibrium

1 =

2 =

3

Can this two-player society do better? More specifically, is consuming

3clean air for each player too much (or too little)?

— The ‘right way’ to answer this question is using the Pareto princi-ple (Vilfredo Pareto, 1848-1923) — can we find another action profile = (1 2) that will make both players better off than in the Nashequilibrium?

— To this end, the function we seek to maximize is the social welfarefunction given by

(1 2) = 1 + 2 =X2

=1ln() + 2 ln

µ −

X2

=1

¶

— The first-order conditions for this problem are

(1 2)

1=1

1− 2

− 1 − 2= 0

and(1 2)

2=1

2− 2

− 1 − 2= 0

— Solving these two equations simultaneously result the unique Paretooptimal outcome

1 = 2 =

4

The -player Tragedy of the Commons

— In the -player Tragedy of the Commons, the best response of eachplayer = 1 , (−), is given by

(−) = −P

6= 2

— We consider a symmetric Nash equilibrium where each player choosesthe same level of consumption of clean air ∗ (it is subtle to show thatthere cannot be asymmetric Nash equilibria).

— Because the best response must hold for each player and they allchoose the same level then in the symmetric Nash equilibriumall best-response functions reduce to

= −P

6=

2=

− (− 1)

2or

=

+ 1

Hence, the sum of clean air consumed by the firms is

+ 1, which

increases with as Hardin conjectured.

What is the socially optimal outcome with players? And how does societysize affect this outcome?

— With players, the social welfare function given by

(1 ) =X

=1

=X

=1ln() + ln

³ −

X

=1´

And the first-order conditions for the problem of maximizing thisfunction are

(1 )

=1

−

−P=1

= 0

for = 1 .

— Just as for the analysis of the Nash equilibrium with players, the solu-tion here is also symmetric. Therefore, the Pareto optimal consumptionof each player can be found using the following equation:

1

−

− = 0

or

=

2

and thus the Pareto optimal consumption of air is equal

2, for any

society size . for = 1 .

Finally, we show there is no asymmetric equilibrium.

— To this end, assume there are two players, and , choosing two dif-ferent 6= in equilibrium.

— Because we assume a Nash equilibrium the best-response functions of and must hold simultaneously, that is

= − ̄ −

2and =

− ̄ − 2

where ̄ be the sum of equilibrium choices of all other players except and .

— However, if we solve the best-response functions of players and

simultaneously, we find that

= = − ̄

3

contracting the assumption we started with that 6= .

Oligopolistic competition

• Oligopoly is a form of market structure is — a market in which only a fewfirms compete with one another, and entry of new firms is impeded.

• The situation is known as the Cournot model after Antoine AugustinCournot, a French economist, philosopher and mathematician (1801-1877).

• In the basic example, a single good is produced by two firms (the industryis a “duopoly”).

Cournot’s oligopoly model (1838) (Antoine Augustin Cournot, an econo-mist, philosopher and mathematician, 1801-1877).

— A single good is produced by two firms (the industry is a “duopoly”).

— The cost for firm = 1 2 for producing units of the good is givenby (“unit cost” is constant equal to 0).

— If the firms’ total output is = 1 + 2 then the market price is

= −

if ≥ and zero otherwise (linear inverse demand function). Wealso assume that .

To find the Nash equilibria of the Cournot’s game, we can use the proce-dures based on the firms’ best response functions.

But first we need the firms payoffs (profits):

1 = 1 − 11= (−)1 − 11= (− 1 − 2)1 − 11= (− 1 − 2 − 1)1

and similarly,

2 = (− 1 − 2 − 2)2

Firm 1’s profit as a function of its output (given firm 2’s output) 

Profit 1

Output 1 2

21 qcA 2

'21 qcA

22' qq

2q

To find firm 1’s best response to any given output 2 of firm 2, we needto study firm 1’s profit as a function of its output 1 for given values of2.

Using calculus, we set the derivative of firm 1’s profit with respect to 1equal to zero and solve for 1:

1 =1

2(− 2 − 1)

We conclude that the best response of firm 1 to the output 2 of firm 2

depends on the values of 2 and 1.

Because firm 2’s cost function is 2 6= 1, its best response function isgiven by

2 =1

2(− 1 − 2)

A Nash equilibrium of the Cournot’s game is a pair (∗1 ∗2) of outputs

such that ∗1 is a best response to ∗2 and

∗2 is a best response to

∗1.

From the figure below, we see that there is exactly one such pair of outputs

∗1 =+2−21

3 and ∗2 =+1−22

3

which is the solution to the two equations above.

The best response functions in the Cournot's duopoly game 

Output 2

Output 1

1cA

21cA

2cA

22cA

)( 21 qBR

)( 12 qBR

Nash equilibrium

Nash equilibrium comparative statics (a decrease in the cost of firm 2) 

A question: what happens when consumers are willing to pay more (A increases)?

Output 2

Output 1

1cA

2cA

22cA

)( 21 qBR

)( 12 qBR

Nash equilibrium I

Nash equilibrium II

21cA

In summary, this simple Cournot’s duopoly game has a unique Nash equi-librium.

Two economically important properties of the Nash equilibrium are (toeconomic regulatory agencies):

[1] The relation between the firms’ equilibrium profits and the profit theycould make if they act collusively.

[2] The relation between the equilibrium profits and the number of firms.

[1] Collusive outcomes: in the Cournot’s duopoly game, there is a pair of out-puts at which both firms’ profits exceed their levels in a Nash equilibrium.

[2] Competition: The price at the Nash equilibrium if the two firms have thesame unit cost 1 = 2 = is given by

∗ = − ∗1 − ∗2

=1

3(+ 2)

which is above the unit cost . But as the number of firm increases, theequilibrium price deceases, approaching (zero profits).

Cournot’s oligopoly game (many firms)

— Suppose all firms have the same unit cost, i.e. = for all firms .Firm 1’s payoff (profit) is given by

1 = 1 − 1

= (−)1 − 1

= (− 1 − 2 − − )1 − 1

= (−P=1 − )1

— To find firm 1’s best response to any given outputs 2 of theother firms, we need to study firm 1’s profit as a function of its output1 for given values of 2 .

— Thus firm 1’s best response function is

1 =1

2(− 2 − − − )

— The best response functions of every other firm is the same so theconditions for (∗1

∗2

∗) to be a Nash equilibrium are

∗1 = 1(∗−1)

...∗ = 1(

∗−)

where ∗− stands for the list of the outputs of all the firms except firm.

— Let the common value of the firms’ outputs in the (unique symmetric)Nash equilibrium be ∗. Then each best response function is

∗ =1

2(− (− 1)∗ − )

Rearranging,

− (+ 1)∗ − = 0

or

∗ =−

+ 1

— The price at this equilibrium is

−

+ 1(− )

so as the number of firms increases this price decreases, approaching as →∞ (increases without bound).

Stackelberg’s duopoly model (1934)

How do the conclusions of the Cournot’s duopoly game change when thefirms move sequentially? Is a firm better off moving before or after theother firm?

Suppose that 1 = 2 = and that firm 1 moves at the start of the game.We may use backward induction to find the subgame perfect equilibrium.

— First, for any output 1 of firm 1, we find the output 2 of firm 2

that maximizes its profit. Next, we find the output 1 of firm 1 thatmaximizes its profit, given the strategy of firm 2.

Firm 2

Since firm 2 moves after firm 1, a strategy of firm 2 is a function thatassociate an output 2 for firm 2 for each possible output 1 of firm 1.

We found that under the assumptions of the Cournot’s duopoly game Firm2 has a unique best response to each output 1 of firm 1, given by

2 =1

2(− 1 − )

(Recall that 1 = 2 = ).

Firm 1

Firm 1’s strategy is the output 1 the maximizes

1 = (− 1 − 2 − )1 subject to 2 =12(− 1 − )

Thus, firm 1 maximizes

1 = (− 1 − (1

2(− 1 − ))− )1 =

1

21(− 1 − )

This function is quadratic in 1 that is zero when 1 = 0 and when1 = − . Thus its maximizer is

∗1 =1

2(− )

Firm 1’s (first‐mover) profit in Stackelberg's duopoly game 

Profit 1

Output 1 2

1cA cA

)(21

111 cqAq

We conclude that Stackelberg’s duopoly game has a unique subgame per-fect equilibrium, in which firm 1’s strategy is the output

∗1 =1

2(− )

and firm 2’s output is

∗2 =1

2(− ∗1 − )

=1

2(− 1

2(− )− )

=1

4(− )

By contrast, in the unique Nash equilibrium of the Cournot’s duopoly game

under the same assumptions (1 = 2 = ), each firm produces1

3(− ).

The subgame perfect equilibrium of Stackelberg's duopoly game 

Output 2

Output 1 3

cA cA

2cA )( 12 qBR

Nash equilibrium (Cournot)

2cA

Subgame perfect equilibrium (Stackelberg)

Bertrand’s oligopoly model (1883)

In Cournot’s game, each firm chooses an output, and the price is deter-mined by the market demand in relation to the total output produced.

An alternative model, suggested by Bertrand, assumes that each firmchooses a price, and produces enough output to meet the demand it faces,given the prices chosen by all the firms.

=⇒ As we shell see, some of the answers it gives are different from the answersof Cournot.

Suppose again that there are two firms (the industry is a “duopoly”) andthat the cost for firm = 1 2 for producing units of the good is givenby (equal constant “unit cost”).

Assume that the demand function (rather than the inverse demand functionas we did for the Cournot’s game) is

() = −

for ≥ and zero otherwise, and that (the demand function inPR 12.3 is different).

Because the cost of producing each until is the same, equal to , firm

makes the profit of − on every unit it sells. Thus its profit is

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩( − )(− ) if 1

2( − )(− ) if =

0 if

where is the other firm.

In Bertrand’s game we can easily argue as follows: (1 2) = ( ) is theunique Nash equilibrium.

Using intuition,

— If one firm charges the price , then the other firm can do no betterthan charge the price .

— If 1 and 2 , then each firm can increase its profit bylowering its price slightly below .

=⇒ In Cournot’s game, the market price decreases toward as the number offirms increases, whereas in Bertrand’s game it is (so profits are zero)even if there are only two firms (but the price remains when the numberof firm increases).

Avoiding the Bertrand trap

If you are in a situation satisfying the following assumptions, then you willend up in a Bertrand trap (zero profits):

[1] Homogenous products

[2] Consumers know all firm prices

[3] No switching costs

[4] No cost advantages

[5] No capacity constraints

[6] No future considerations

Auctions

Types of auctions

Sequential / simultaneous

Bids may be called out sequentially or may be submitted simultaneouslyin sealed envelopes:

— English (or oral) — the seller actively solicits progressively higher bidsand the item is soled to the highest bidder.

— Dutch — the seller begins by offering units at a “high” price and reducesit until all units are soled.

— Sealed-bid — all bids are made simultaneously, and the item is sold tothe highest bidder.

First-price / second-price

The price paid may be the highest bid or some other price:

— First-price — the bidder who submits the highest bid wins and pay aprice equal to her bid.

— Second-prices — the bidder who submits the highest bid wins and paya price equal to the second highest bid.

Variants: all-pay (lobbying), discriminatory, uniform, Vickrey (WilliamVickrey, Nobel Laureate 1996), and more.

Private-value / common-value

Bidders can be certain or uncertain about each other’s valuation:

— In private-value auctions, valuations differ among bidders, and eachbidder is certain of her own valuation and can be certain or uncertainof every other bidder’s valuation.

— In common-value auctions, all bidders have the same valuation, butbidders do not know this value precisely and their estimates of it vary.

First-price auction (with perfect information)

To define the game precisely, denote by the value that bidder attachesto the object. If she obtains the object at price then her payoff is −.

Assume that bidders’ valuations are all different and all positive. Numberthe bidders 1 through in such a way that

1 2 · · · 0

Each bidder submits a (sealed) bid . If bidder obtains the object, shereceives a payoff − . Otherwise, her payoff is zero.

Tie-breaking — if two or more bidders are in a tie for the highest bid, thewinner is the bidder with the highest valuation.

In summary, a first-price sealed-bid auction with perfect information is thefollowing strategic game:

— Players: the bidders.

— Actions: the set of possible bids of each player (nonnegative num-bers).

— Payoffs: the preferences of player are given by

=

( − ̄ if = ̄ and if = ̄0 if ̄

where ̄ is the highest bid.

The set of Nash equilibria is the set of profiles (1 ) of bids with thefollowing properties:

[1] 2 ≤ 1 ≤ 1[2] ≤ 1 for all 6= 1[3] = 1 for some 6= 1

It is easy to verify that all these profiles are Nash equilibria. It is harderto show that there are no other equilibria. We can easily argue, however,that there is no equilibrium in which player 1 does not obtain the object.

=⇒ The first-price sealed-bid auction is socially efficient, but does not neces-sarily raise the most revenues.

Second-price auction (with perfect information)

A second-price sealed-bid auction with perfect information is the followingstrategic game:

— Players: the bidders.

— Actions: the set of possible bids of each player (nonnegative num-bers).

— Payoffs: the preferences of player are given by

=

( − ̄ if ̄ or = ̄ and if = ̄0 if ̄

where ̄ is the highest bid submitted by a player other than .

First note that for any player the bid = is a (weakly) dominantaction (a “truthful” bid), in contrast to the first-price auction.

The second-price auction has many equilibria, but the equilibrium = for all is distinguished by the fact that every player’s action dominatesall other actions.

Another equilibrium in which player 6= 1 obtains the good is that inwhich

[1] 1 and 1[2] = 0 for all 6= {1 }

Common-value auctions and the winner’s curse

Suppose we all participate in a sealed-bid auction for a jar of coins. Onceyou have estimated the amount of money in the jar, what are your biddingstrategies in first- and second-price auctions?

The winning bidder is likely to be the bidder with the largest positive error(the largest overestimate).

In this case, the winner has fallen prey to the so-called the winner’s curse.Auctions where the winner’s curse is significant are oil fields, spectrumauctions, pay per click, and more.

First-price auction class experiment

0.0

5.1

.15

.2.2

5Fr

actio

n

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Bid

Second-price auction class experiment

0.0

5.1

.15

.2.2

5Fr

actio

n

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Bid