game theory homework
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course_6_gt.pptxGame Theory Midterm undergraduate
Graduate
1. Give examples of games with (a) one Nash equilibrium, (b) two Nash equilibria, and (c) three Nash equilibria.
2. Suppose that we have two (duopoly) firms and they each own a homogeneous resource, such as a forest or a fishing pool, of size 25. Each of them can withdraw any nonnegative amount, c or d, in the first period, provided of course that c ≤ 25, d ≤ 25. There are two periods, and whatever is not extracted in the first period is extracted in the second. Finally, the extracted amounts are sold in a (common) market whose demand curve is given by p = 100-Q, where Q is the total amount extracted in any period. Assume that each firm is a profit maximizer and that costs of production are zero.
a. Write down the strategic form of this game.
b. What is the best response for firm 1 if it thinks that firm 2 will produce 7.5 units of output? What is the best response function of firm 1? (In answering this question, you will need to use the facts that profits are maximized when the slope of the profit function is zero and that the slope of a quadratic function a + bx – cx2 is b – 2cx).
c. What is the Nash equilibrium extraction levels in period 1? period 2? What about market price?
3. The next few questions return to the analysis of the model of price competition with a market whose demand curve is Q = 6 – p where p is the lower of the two prices. The lower priced firm gets the entire market, and if the two firms post the same price, then each gets half the market. Suppose that prices can only be quoted in dollar units and costs of production are zero. Suppose, finally, that price competition continues indefinitely; that is, every time the two firms compete they think that there is a probability δ that they will compete again.
c. Consider the following strategy: Price at 2 dollars each and continue with that price if it has been maintained by both firms in the past. Otherwise, switch to a price of a dollar. For what values of δ is this strategy a subgame perfect equilibrium? Explain.
d. Show that there is also a subgame perfect equilibrium in which the price is always 2 dollars but which is sustained by a forgiving trigger. Be explicit about the nature of the forgiving trigger.
e. Suppose that δ = 0.9. What is the maximum price that can arise in a subgame perfect equilibrium of this model? Explain.
4. Analyze a generalization of the Stackelberg model for n+1 firms. What kind of equilibrium we find in this situation? Analyze the model.
5 Present and discuss North’s view of the development of Champagne fairs in medieval France using game theory.
Game Theory
Course 6
An Application: OPEC
OPEC is an organization of oil producers; it was formed in September 1960 at the primary urging of the bigger producers Saudi Arabia, Iran, and Venezuela. OPEC has 13 member states.
In the first half of the 20th century the primary producers and exporters of oil were the United States and Venezuela and in later years in the Soviet Union.
Two dramatic changes occurred during and after World War II.
First, new sources of oil were discovered in the Middle East, and production capacity was greatly increased in Iran and Iraq (for the Allied war effort).
Second, the postwar industrial boom in the United States increased demand several fold. Indeed the United States went from being a net exporter to an importer of oil (by 1970, 60% of U.S. oil demand was being met by imports).
The price history of oil
Phase 1, Before 1960: Prices were both low and stable. For example, the price of a barrel of oil only rose from $1.25 in 1950 to $1.75 in 1960.
Phase 2, 1960 to October 1973: Prices remained low but began to creep up. Through the 1960s they remained in a one-dollar band between $1.50 and $2.50and by the middle of 1973 they were up to $5.
Phase 3, October 1973 to 1979: Prices were both high and stable. The most dramatic phase was undoubtedly the mid-1970s; the price of a barrel of oil went from $5 to $17 in the last two months of 1973. It remained in the twenties throughout the decade.
Phase 4, 1980 Onwards: Prices have been lower and unstable; that is, there has been a lot of volatility. For example, prices were as high as $30 a barrel around 1982 and as low as $10 a barrel in the early 1990s.
A Simple Model of the Oil Market
Oil-producing nations compete with each other as Cournot-style competitors in the world oil market.
There is a very active worldwide spot market as well as a market in oil futures. The main markets are the International Petroleum Exchange (IPE) based in London and the New York Mercantile Exchange (NYME).
Simple stage game
There are two oil producers, say, Saudi Arabia (SA) and Venezuela (VA).
Each of these producers can produce either a high output or a low output.
SA and VA are different-sized producers; let us suppose, therefore, that the two output levels for SA are QH = 10 mbd and QL = 8 mbd; for VA these output levels are qH = 7 mbd and qL = 5 mbd. Aggregate output can therefore be any one of three levels; when both producers withhold output the total is 13 mbd, when only one withholds it is 15 mbd and when both overproduce it is 17 mbd.
Good Times…
On the demand side, let us suppose that demand conditions can be either good or bad.
When conditions are good, a total output of 13 mbd fetches a price of $25 per barrel,
Whereas the price is only $22 per barrel for an aggregate output of 15 mbd
The price is just $19 for the highest output level.
Finally, suppose that the marginal cost of production is $5 a barrel.
Pay-off matrix for Good times
SA \ VA. qL qH
QL 160, 100 136, 119
QH 170, 85 140, 98
Bad Times…
In contrast, when times are bad, that is, when demand is weak suppose the three prices are respectively $16, $15, and $14 per barrel for high, medium, and low total output.
Suppose also that the costs of production are no different.
Pay-off matrix for Bad times
SA \ VA qL qH
QL 88, 55 80, 70
QH 100, 50 90, 63
Phase 1, Before 1960
The 1950s was the first decade of significant growth in oil demand. Yet the rates of growth were still far below what was to come in the 1960s and 1970s. The 1950s was also a period of low prices.
One explanation of the 1950s, then, is that demand at that time can be characterized as a bad or weak demand. In the stage game of Bad Times there is exactly one (dominant strategy) equilibrium; that is, each producer produces a high output. Saudi Arabia produces 10 mbd, Venezuela produces 7 mbd, and consequently the price is the low price of $14 per barrel. The associated profits are (90, 63).
Phase 2, 1960-October 1973: The 1960s witnessed a continuing increase in demand; demand was significantly higher in this decade than in the 1950s, although per capita consumption was still below the levels that would be witnessed in the 1970s.
We will model these two observations in the following way: Suppose that in each period there is a chance
that demand is robust, with payoffs given by Good Times. The probability that demand is robust in any given period will be denoted p; with the remaining probability, (1 - p), profits are given by Bad Times matrix.
When demand is robust we are in a true Prisoners' Dilemma situation. Although joint profits are maximized at (QL, qL), the dominant strategy in the stage game is (QH, qH).
Can OPEC maintain high prices in good years while continuing to target a low price in bad demand years?
They were not able to hold up prices even in good years during the 1960s.
Pay-off matrix for Good times
SA \ VA. qL qH
QL 160, 100 136, 119
QH 170, 85 140, 98
Phase 3, October 1973-1979: Things change in the early 1970s. Demand peaks around this time. A shorthand modeling of this condition is to imagine that demand is almost never anemic during this phase, that is, that p is (virtually) equal to 1.
Grim trigger strategy: production is low to begin with and remains low provided that honoring the quota remains the observed pumping pattern; if either producer starts to exceed its quota, this cooperation breaks down and each starts producing at capacity forever thereafter.
Cooperating on withholding output yields Saudi Arabia a profit stream of
160+160δ+160δ2 ….
whereas overproduction in any period yields an immediate increase of profits to 170 but is followed thereafter by the punishment of a grim trigger, which yields a profit stream of:
170+140δ+140δ2 ….
it pays not to cheat against OPEC's high-price policy if δ>1/3.
Cooperation yields Venezuela a profit stream of:
100+100δ+100δ2 ….
while overproduction in any one period yields
119+98δ+98δ2 ….
Venezuela will refrain from overproducing only if δ>19/21 (more prone to cheating)
Phase 4, 1980 Onward: This has been a period of unstable prices driven by demand uncertainty. Demand fluctuates in part because of conservation efforts in industrialized countries and increasing reliance on alternative energy sources. There have also been discoveries of non-OPEC oil supplies, such as from the North Sea. In other words, the probability of robust demand, p, is again less than 1.
Repeated Games with Demand Uncertainty
In many economic applications, such as competition within OPEC, the game that is played in any one year is typically a little different from the one played in the previous year.
The question that we will discuss now is how to modify the analysis to take account of such variations in market conditions.
In any stage, each producer can now make any one of four decisions:
withhold production regardless of demand conditions,
withhold production only if demand conditions are good,
withhold only if they are bad,
overproduce no matter what.
In other words, SA has four strategies within a stage game: (QL, QL), (QL, QH), (QH, QL), and (QH, QH).
In each case, interpret the first component as SA's choice if demand is good, while the second component is the choice if demand is bad.
Likewise, VA has four analogous strategies within a stage game.
Denoting by p the probability that demand is going to be robust, the expected profit to each producer in any given period is given by the following payoff matrix.
SA \ VA qL, qL qL, qH etc.
QL, QL 160p + 88(1 - p), 100p + 55(1 - p) 160p + 80(1 - p), 100p + 70(1 - p)
QL, QH 160p + 100(1 - p), 100p + 50(1 - p) 160p + 90(1 - p), 100p + 63(1 - p)
QH, QL 170p + 88(1 - p), 85p + 55(1 - p) 170p + 80(1 - p), 85p + 70(1 - p)
QH,QH 170p + 100(1 - p), 85p + 50(1 - p) 170p + 90(1 - p), 85p + 63(1 - p)
Show that the total expected profits are maximized if the two producers both produce low in good demand periods and high in bad demand periods.
Can OPEC sustain this profit maximizing production policy?
Again suppose that the "punishment" for cheating on OPEC is the following grim trigger: overproduction no matter what the market demand conditions are like, forever after.
Following the cartel's best policy L output in
good years, H in bad, SA gets a future profit stream:
[160p+90(1-p)]δ+[160p+90(1-p)]δ2 ….
For good year, 160 + future rev. A deviation is 170 + future punishment
[140p+90(1-p)]δ+[140p+90(1-p)]δ2 ….
Is the present gain of 10 worth future punishment? If δ>1/(1+2p), NO
Note the two patterns:
First, the higher is p, the more effective is OPEC (the bigger is the range of δ over which neither producer cheats). This conclusion should be intuitive; when demand conditions are bad, OPEC quotas are completely unnecessary, and ineffective anyway, because withholding output actually lowers total profits.
It is only when demand is good that it is more attractive for each producer to cheat on his quota and increase his own profits at the expense of the cartel. Hence, cheating needs to be deterred by the threat of "flooding the market" in the future.
Second, VA, the smaller producer, is more likely to cheat on OPEC (the critical δ is always higher for VA).
This conclusion is also intuitive; the smaller producer has more to gain today by overproducing. After all, the only thing to worry about is that overproduction will depress the price in the current market, and that is less of a worry if current production is small.
The thesis of phases 2 and 4
In the 1960s, phase 2, demand was growing but not sufficiently quickly; that is, p was low. In particular, not all members of OPEC had an incentive to sustain a cartel. Hence, OPEC was unable to maintain high prices even in good demand years.
However, starting in the early 1980s, although demand dropped off from the early 1970s peak, it nevertheless has been sufficiently high. What we see, therefore, is high prices in good demand years and low prices in bad demand years.
Homework 3
Extend the model that we just discussed in one direction:
Number of countries
Level of production
Cost of production
OPEC's oil reserves are shrinking over time. This shrinkage will eventually have an impact on the profitability of any production policy.
The role of non-OPEC production.
Questions
How does the resource stock yt evolve over time, and is there an eventual size that can be sustained?
What is the socially optimal sustainable resource stock?
Does strategic interaction lead to over extraction of the resource?
Social Optimum
We need to consider the sum of the two players' utilities and maximize it
Suppose, that there are exactly two periods.
Backward induction => last period
Max ln c1 + ln c2 st c1 + c2 < or = y
The Commons Problem: A Model
Resource stock, yt >0
2 players
Player i’s consumption/extraction – cit
Constraint: yt ≥ c1t +c2t
yt – (c1t +c2t )= xt – investment for future growth
We consider a renewable resource yt+1 > xt
utility of c is ln (c ) – larger consumption brings lower utility gains
Production function: yt+1 =10 (xt )^(1/2) – larger stock brings lower gains
If maximize utility => nothing left so c1+c2=y. Then
Max ln c1 + ln (y-c1) wrt c1.
FOC => c1=c2=y/2
The social optimal utility for each player is V1(y)=ln y –ln 2. For 2 periods, the max. problem:
Max ln c1+ln c2+2δV1[10(y-c1-c2)0.5 ] st c1+c2< or = y
We can write the problem as:
Max ln c1 + ln c2 + δ ln(y-c1-c2)
FOC for c1 and c2 => same results c1=c2=c and c=y/(2+δ) => less than last period, some left for investment.
The socially optimal per capita consumption is V2=(1+δ/2)ln y +constants
For a 3 periods, in the first period the problem is:
Max ln c1+ln c2+2δV2(10(y-c1-c2)0.5 ] st c1+c2< or = y.
FOC => c1=c2=y/2(1+δ/2+δ2 /4)-1
By induction, for T periods, c=1/[2(1+δ/2+δ2 /4+…+ (δ/2)T-1 )]
For infinitely repeated game c=(1-δ/2)y/2
For δ=0.8, each player consumes 30% of available stock and 40% are investment for the next period.
Strategic approach
Again using backward induction:
In the last period, both players consume everything – y/2, with utility W1(y)=ln (y/2)= ln y + const.
In the first period, the problem for player 1 is:
Max ln c1+δW1[10(y-c1-θy)0.5 st c1 < por = (1-θ)y where θ is the fraction of resources that player 2 is expected to consume in the first period.
Best response to θ is (1-θ)/(1+δ/2). Due to the symmetry of the problem, we look at a symmetric solution: θ = (1-θ)/(1+δ/2).
Using a similar approach as for social optimum, we find c*(y)=(1-δ/2)/(2-δ/2)y.
With δ=0.8, each player consumes 37.5% of stock with 25% as investment for the next period.
The general lesson is that unilateral extraction leads to over extraction; consumption in the equilibrium solution is higher than in the socially optimal solution.
The intuition for this conclusion is precisely that in the equilibrium solution a player only collects a part of his action's consequences.
Another consequence of this argument is that the sustainable socially optimal stock is higher than the achievable equilibrium stock. We have already seen this outcome with some numbers, but here is the general argument: A sustainable stock is one that keeps getting regenerated
