This course provides a brief introduction to game theory. Our main goal is to understand the basic ideas behind the key concepts in game theory, such as equilibrium, rationality, and cooperation. The course uses very little mathematics, and it is ideal for those who are looking for a conceptual introduction to game theory. Business competition, political campaigns, the struggle for existence by animals and plants, and so on, can all be regarded as a kind of “game,” in which individuals try to do their best against others. Game theory provides a general framework to describe and analyze how individuals behave in such “strategic” situations. This course focuses on the key concepts in game theory, and attempts to outline the informal basic ideas that are often hidden behind mathematical definitions. Game theory has been applied to a number of disciplines, including economics, political science, psychology, sociology, biology, and computer science. Therefore, a warm welcome is extended to audiences from all fields who are interested in what game theory is all about.
4-6 Cooperation of gas Stations in Long-Term Relationship Part I: Need For Cooperation
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来自 The University of Tokyo 的课程
Welcome to Game Theory
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从本节课中
Sustaining Cooperation
The final module is devoted to the most important and most general message of game theory: rational behavior quite often leads to a socially undesirable outcome. We will first try to understand the basic reason for this, and then see how this insight of game theory has made fundamental impacts in the natural and social sciences. Finally, we will learn some general methods to overcome this problem.
与讲师见面
Michihiro KandoriProfessor
Faculty of Economics
So now let me explain how a long-term relationship works to sustain cooperation
and a simple example is two gas stations.
Okay so my lecture comes in two parts.
In this lecture I explain the need for cooperation between gas stations.
And in the next lecture, I'm going to talk about how to sustain cooperation
in a long term contract of two gas stations.
Okay well in a crowded city like Tokyo
oftentimes gas stations are located in very nearby locations.
And, in the extreme case, I like this picture a lot,
two gas stations are located right next to each other.
So let's examine what's going to happen in this situation.
Okay so, every morning those gas stations post prices, right?
3, $3.5 per gallon, and so on.
And they are selling identical product, gasoline, 'kay.
So what's the implication?
Well, if one gas station is cheaper than the other,
all customers rush to the cheaper one.
Okay, different prices means, that all customers go to the cheaper station, okay?
Because gasoline is gasoline.
Okay, if they charge the same price well I assume that they split customers equally.
So this is a consequence of selling identical product at the same pla, place.
What about the cost of selling the gasoline?
What's the unit cost of selling one gallon, gallon of gasoline?
Well, I assume that it's very likely that they have the same constant unit cost.
So unit cost is the wholesale price
which those gas station pay to a gasoline company.
And their labor costs and so on.
So let's assume that they have the identical same unit identical and
constant unit cost for selling gasoline, okay?
So the, the situation is summarized by this diagram.
So, horizontal axis, I measure the quantity of gasoline, and
vertical axis measures the price.
And the demand for gasoline is described by this downwards sloping demand curve.
And let's say unit cost of sale is $2 per gallon, okay?
So what could be best outcome for those two gas stations?
Well, they can charge a price that maximizes their total profit.
So remember that this is a total demand for those two gas stations.
And say, if they charge $3 each, then the demand is given by this demand curve.
And for one unit of gasoline, the profit is a difference between $3,
the price, and $2 cost, 'kay?
For each unit the profit is $1.
And this is the quantity solved, which is determined by market demand.
And this gray area here represents total profit, 'kay?
So if you change the price from $3 then you change the, the size of this area.
But you choose the price so
that you maximize the size of this area, total profit.
And let's assume that this total profit, this gray area,
is maximized when the price is 3, okay?
So at the best outcome for those two gas stations,
they should charge $3 each to maximize their total profit.
And since they are charging exactly the same price,
$3, they split customers equally, okay?
So this is gas station 1's profit and this is gas station 2's profit.
They are splitting total maximized profit equally.
This is a best point for those two gas stations.
Okay, is this best outcome sustainable for them?
Unfortunately, it is not.
Because, for example,
gas station 1 can undercut supplies slightly, you know, 2.99.
And then all the customer rush to gas station 1, 'kay.
So by slightly undercutting the price
gas station 1 can steal all customers from station 2.
And, since the price is not changing so much, effectively
gas station 1 is stealing all the profit of, from 2, gas station 2.
Okay, so by slightly undercutting the price from optimal price of $3,
gas station 1 can almost double its profit.
So each gas station is tempted to undercut the price to increase, or
steal the profit from the other gas station.
Okay, so the best outcome is this one.
They charge the max, optimal price that maximizes their total profit,
$3, and they split customers equally.
This is a happy situation, optimal situation for the society, but
unfortunately it's unstable.
Each gas station has an incentive to undercut supplies and
to steal customers from the other station.
So in game theory terminology this best outcome is not a Nash equilibrium.
Each player has an incentive to deviate.
Well, you can see that even if the price is not optimal, say $2.5,
then the same logic applies.
Each company, or each gas station can undercut the price and
steal the other station's profit.
The only Nash equilibrium in this situation is that they charge
the price which is equal to the cost, and they are both earning zero profit, okay?
Obviously, this is a Nash equilibrium because
each player cannot possibly gain any positive profit.
If other company, or if other gas station, if is charging price too,
and if you increase your price, nobody comes to you, okay?
So your profit remains to be zero.
And if you undercut the price, all customers come to you.
That's good for you.
But, now your price is below the cost.
So you lose.
Your profit is negative.
So given that the other company is ju,
the other gas station is charging a price which is equal to cost, then no
matter how you change your price you cannot possibly gain any positive profit.
So this situation no one can gain by deviating alone, and
this is the Nash equilibrium of price competition game.
Is it clear?
Okay, again, in this gas station example what is good for the society,
total profit maximization, is not equal to what is good for each individual.
That is Nash equilibrium.
Okay, so let's go back to the reality, okay.
So those two gas stations are happily operating for a long time.
But the Nash equilibrium of price competition game predicts zero profit for
those companies.
Okay, but in reality, those gas stations are enjoying positive profit obviously.
So what's wrong with the game theoretic explanation?
Well this prediction here by the game theory, they should only earn zero profit.
This prediction should be true when
they play the price competition game only once, okay?
So they post the price today, and they get some customers today, and
that's the end of the game.
No, they go out of business by the end of today.
And if that's the situation, this prediction should be so,
should be an accurate description of what is happening in reality.
But the important point you should notice is that
those two stations are not playing this price competition game only once, 'kay?
They are located side-by-side, and they play this game of price competition
again and again, today, tomorrow, the day after tomorrow, 'kay.
In reality, they play the price competition game every day, and
that's what I call long-term relationship, okay?
And in the long-term relationship a positive profit may be achieved.
That's going to I'm going to explain how this happens in the next lecture.
But possibility of earning positive profit is coming from long-berm relationship.
Okay.
So let's think about how to formalize this situation,
long-term relationship of two gas stations, okay?
Well, you have the same players, gas stations 1 and 2.
And they play the same game, the price competition game, every day.
Okay.
So this situation is formalized by what is called repeated game.
Repeated game has the same set of players, playing the same game again and
again, okay?
By analyzing this dynamic game, instead of analyzing price competition in one day,
we can see that those gas stations can actually sustain positive profit.
Okay? So a ga,
repeated game is a model of long-term relationship, which shows, you know,
rational players can sustain cooperation in long-term relationship.
Okay, so let me just explain one terminology.
So the whole dynamic game played over time, today, tomorrow,
the day after tomorrow, this dynamic interaction is called repeated game.
And price competition game is played every day.
The game played every day is called the stage game of a repeated game, okay.
So let me briefly explain the intuition about how
they can sustain high price in dynamic price competition, repeated game.
Well the following could be a very plausible considerations
of those two gas stations, okay?
Well, I can undercut the price today, and I can certainly increase
my profit today by stealing customers from the other gas station.
That's true, okay?
But if I do that, if I undercut the price today,
then it may trigger very harsh or cutthroat price competition in the future.
Other you know, gas station,
may react and price in the future may, may go down, okay?
And if that happens, after all, that's not going to be good for
me, so therefore, even though I can gain something today,
then future, you know, consequence is really bad.
So overall, you know, I'm better off by refraining from undercutting the price.
This is a very plausible consideration.
And in the next I'm going to show you how to formalize this idea.
