UC Berkeley Department 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) on which they are each entitled to let their cows graze. If a herder put more than his allotted number of cattle on the common, overgrazing could result.
— Each additional animal has a positive effect for its herder, but the cost of 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 Tragedy of the Commons,” published in the journal Science. The essay derived its 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 human population has increased, the commons has had to be abandoned in one aspect after another.” — “The only way we can preserve and nurture other and more precious freedoms 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 to breed. Only so, can we put an end to this aspect of the tragedy of the commons.”
“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 our planet.
— There is amount of clean air, and any consumption of clean air comes out of this common resource. Each player =1 chooses his consumption of clean air for production 0 and the amount of ≥ clean air left is therefore − =1 X — The benefit of consuming an amount 0 of clean air gives player ≥ abenefit equal to ln(). Each player also enjoys consuming the reminder of the clean air, giving each a benefit ln − =1 ³ X ´ — Hence, the value for each player from the action profile (outcome) =(1 ) is give by ( )=ln()+ln − − =1 µ X ¶ — To get player ’s best-response function, we write down the first-order condition of his payoff function:
( ) 1 1 − = =0 − − =1 and thus P = ( )= − 6 − P2 Thetwo-playerTragedyoftheCommons
— To find the Nash equilibrium, there are equations with unknown that need to be solved. We first solve the equilibrium for two players. Letting () be the best response of player ,wehavetwobest- response functions:
2 1 ( )= − and ( )= − 1 2 2 2 1 2 — If we solve the two best-response functions simultaneously, we find the unique (pure-strategy) Nash equilibrium = = 1 2 3 Can this two-player society do better? More specifically, is consuming 3 cleanairforeachplayertoomuch(ortoolittle)?
— The ‘right way’ to answer this question is using the Pareto princi- ple (Vilfredo Pareto, 1848-1923) — can we find another action profile =(12) that will make both players better off than in the Nash equilibrium?
— To this end, the function we seek to maximize is the social welfare function given by 2 2 (12)=1 + 2 = ln( )+2ln =1 − =1 X µ X ¶ — The first-order conditions for this problem are ( ) 1 2 1 2 = =0 1 1 − 1 2 − − and ( ) 1 2 1 2 = =0 2 2 − 1 2 − − — Solving these two equations simultaneously result the unique Pareto optimal outcome = = 1 2 4 The -player Tragedy of the Commons
— In the -player Tragedy of the Commons, the best response of each player =1 , ( ), is given by − = ( )= − 6 − P2 — We consider a symmetric Nash equilibrium where each player chooses the same level of consumption of clean air ∗ (itissubtletoshowthat there cannot be asymmetric Nash equilibria). — Because the best response must hold for each player and they all choose the same level then in the symmetric Nash equilibrium all best-response functions reduce to = ( 1) = − 6 = − − P 2 2 or = +1 Hence, the sum of clean air consumed by the firms is ,which +1 increases with as Hardin conjectured. What is the socially optimal outcome with players? And how does society size affect this outcome?
— With players, the social welfare function given by ( )= 1 =1 = X ln( )+ ln =1 − =1 And the first-order conditionsX for the problem³ ofX maximizing´ this function are
( ) 1 1 = =0 − − =1 for =1 . P — Just as for the analysis of the Nash equilibrium with players, the solu- tion here is also symmetric. Therefore, the Pareto optimal consumption of each player can be found using the following equation: 1 =0 − or − = 2 and thus the Pareto optimal consumption of air is equal ,forany 2 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 = in equilibrium. 6 — Because we assume a Nash equilibrium the best-response functions of and must hold simultaneously, that is
¯ ¯ = − − and = − − 2 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 few • firms compete with one another, and entry of new firmsisimpeded.
The situation is known as the Cournot model after Antoine Augustin • Cournot, a French economist, philosopher and mathematician (1801-1877).
In the basic example, a single good is produced by two firms (the industry • is 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 given by (“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). We ≥ also 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
q'2 q2
q2
A c q A c q' Output 1 1 2 1 2 2 2
To find firm 1’s best response to any given output 2 of firm 2, we need to study firm 1’s profit as a function of its output 1 for given values of 2.
Using calculus, we set the derivative of firm 1’s profit with respect to 1 equaltozeroandsolvefor1: 1 1 = ( 2 1) 2 − −
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 = 1, its best response function is 6 given by 1 2 = ( 1 2) 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
+2 21 +1 22 1∗ = 3− and 2∗ = 3− which is the solution to the two equations above. The best response functions in the Cournot's duopoly game Output 2
A c1
BR1(q2 )
Nash equilibrium A c 2 2
BR2 (q1)
A c 1 A c2 Output 1 2
Nash equilibrium comparative statics (a decrease in the cost of firm 2) Output 2
A c1
Nash equilibrium II BR1(q2 )
Nash equilibrium I A c 2 2
BR2 (q1)
A c 1 A c2 Output 1 2 A question: what happens when consumers are willing to pay more (A increases)? In summary, this simple Cournot’s duopoly game has a unique Nash equi- librium.
Two economically important properties of the Nash equilibrium are (to economic regulatory agencies):
[1] The relation between the firms’ equilibrium profits and the profitthey could 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 the same unit cost 1 = 2 = is given by
∗ = 1∗ 2∗ 1 − − = ( +2) 3 which is above the unit cost . But as the number of firm increases, the equilibrium 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 − − − − − =( )1 − =1 − P — To find firm 1’s best response to any given outputs 2 of the other firms, we need to study firm 1’s profit as a function of its output 1 for given values of 2. — Thus firm 1’s best response function is 1 1 = ( 2 ) 2 − − − − — The best response functions of every other firmisthesamesothe conditions 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 = ( ( 1) ) ∗ 2 − − ∗ − 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 the firms move sequentially? Is a firm better off moving before or after the other 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,wefind the output 2 of firm 2 that maximizes its profit. Next, we find the output 1 of firm 1 that maximizes 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 that associate 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 Firm 2 has a unique best response to each output 1 of firm 1, given by 1 2 = ( 1 ) 2 − − (Recall that 1 = 2 = ). Firm 1
Firm 1’s strategy is the output 1 the maximizes 1 1 =( 1 2 )1 subject to 2 = ( 1 ) − − − 2 − − Thus, firm 1 maximizes 1 1 1 =( 1 ( ( 1 )) )1 = 1( 1 ) − − 2 − − − 2 − −
This function is quadratic in 1 that is zero when 1 =0and when 1 = . Thus its maximizer is − 1 = ( ) 1∗ 2 − Firm 1’s (first‐mover) profit in Stackelberg's duopoly game
Profit 1
1 q (A q c) 1 2 1 1
A c A c Output 1 1 2
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 1 = ( ) 2∗ 2 − 1∗ − 1 1 = ( ( ) ) 2 − 2 − − 1 = ( ) 4 − By contrast, in the unique Nash equilibrium of the Cournot’s duopoly game 1 under the same assumptions (1 = 2 = ), each firm produces ( ). 3 − The subgame perfect equilibrium of Stackelberg's duopoly game Output 2
Nash equilibrium (Cournot) A c BR2 (q1) 2 Subgame perfect equilibrium (Stackelberg)
A c A c A c Output 1
3 2
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 firm chooses 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 answers ⇒ of Cournot. Suppose again that there are two firms (the industry is a “duopoly”) and that the cost for firm =1 2 for producing units of the good is given by (equal constant “unit cost”).
Assume that the demand function (rather than the inverse demand function as we did for the Cournot’s game) is
()= − for and zero otherwise, and that (the demand function in ≥ PR 12.3 is different). Because the cost of producing each until is the same, equal to , firm makes the profitof on every unit it sells. Thus its profitis − ( )( ) if − − ⎧ 1 = ( )( ) if = ⎪ 2 − − ⎨⎪ 0 if ⎪ where is the other fi⎩⎪rm.
In Bertrand’s game we can easily argue as follows: (12)=( ) is the unique Nash equilibrium. Using intuition,
— If one firm charges the price , then the other firmcandonobetter than charge the price .
— If 1 and 2 ,theneachfirm can increase its profitby lowering its price slightly below .
= In Cournot’s game, the market price decreases toward as the number of ⇒ firms 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 number of firm increases). Avoiding the Bertrand trap
If you are in a situation satisfying the following assumptions, then you will end 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 simultaneously in sealed envelopes:
— English (or oral) — the seller actively solicits progressively higher bids and the item is soled to the highest bidder.
— Dutch —thesellerbeginsbyoffering units at a “high” price and reduces it until all units are soled.
— Sealed-bid — all bids are made simultaneously, and the item is sold to the 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 a price equal to her bid.
— Second-prices — the bidder who submits the highest bid wins and pay a price equal to the second highest bid.
Variants: all-pay (lobbying), discriminatory, uniform, Vickrey (William Vickrey, 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 each bidder is certain of her own valuation and can be certain or uncertain of every other bidder’s valuation.
— In common-value auctions, all bidders have the same valuation, but bidders 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 attaches to the object. If she obtains the object at price then her payoff is . −
Assume that bidders’ valuations are all different and all positive. Number the bidders 1 through in such a way that
1 2 0 ···
Each bidder submits a (sealed) bid . If bidder obtains the object, she receives a payoff . Otherwise, her payoff is zero. −
Tie-breaking — if two or more bidders are in a tie for the highest bid, the winner is the bidder with the highest valuation. In summary, a first-price sealed-bid auction with perfect information is the following 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 the following properties:
[1] 2 1 1 ≤ ≤ [2] 1 for all =1 ≤ 6 [3] = 1 for some =1 6
It is easy to verify that all these profiles are Nash equilibria. It is harder to 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 following 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 ¯ or = ¯ and if = ¯ = − ¯ ( 0 if where ¯ is the highest bid submitted by a player other than . Firstnotethatforanyplayer the bid = is a (weakly) dominant action (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 dominates all other actions.
Another equilibrium in which player =1obtains the good is that in 6 which [1] 1 and 1 [2] =0for all = 1 6 { } Common-value auctions and the winner’s curse
Suppose we all participate in a sealed-bid auction for a jar of coins. Once you have estimated the amount of money in the jar, what are your bidding strategies 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, spectrum auctions, pay per click, and more. First-price auction class experiment .25 .2 .15 Fraction .1 .05 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Bid
Second-price auction class experiment .25 .2 .15 Fraction .1 .05 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Bid