1-2 Modelling Social Problems as a "Game"

Hi.

So, as I explained in the first lecture game theory formulates many

social problems as a mathematical model of game.

And then apply this illusion concept to get some predictions, okay?

So, in the second lecture I'm going to tell you how you can formulate or

formalize a social problem as a mathematical model called a game.

Okay, so those are the examples of social interaction,

strategic problems, I explained in the first lecture.

And each of them can be formulated as the mathematical,

as the mathematical model of a game.

And in the first lecture, I posed the two questions.

The first question was, what determines the policies of two parties?

The Democrats and the Republicans.

Question number 2.

Okay.

What happens if you construct a bypass from city x to city y?

How it affects the traffic of the existing roads, and

how it affects the traveling time from city x to city y.

Okay.

To answer those questions, you can take intuitive approach.

Okay?

As a human being you have some experience about politics, and

maybe you have been driving your car.

So, in those instances, you,

you can use your intuition to get some answer to those two questions, okay?

In politics, such and such things is going to happen.

And in traffic problem, well, if new bypass comes out,

then, such and such would happen in the existing world.

You can just use your intuition and

your experience to get some answers to those questions.

Those are called, Ad-hoc approach.

So, whenever you get one problem, you use your intuition to get an answer.

If you, f you get out of that problem, then you start,

start over again and you use your intuition to get the answer.

Depending on each problem, you use your intuition and get your answer.

That's what's called ad hoc approach.

In contrast, game theory tried to find,

a unified approach which can be applied to all those social problems, okay?

So what are common, what is common between all those social problems?

Well, those problems share two features.

First individuals try to do their best against others.

Okay.

But they can, they can not do everything.

They have to obey certain rules.

For example, in political struggle between Democrats and

Republicans, there are certain things they can do, okay?

They can design their policy platform, but they cannot really bribe people.

Okay, so every social problem, has a certain set of rules.

So what are, what is common among all those social problems is the individuals

try to do their best against others, under certain set of rules.

Okay, so let's try to formulate social problems which has those two features,

as a simple mathematical model, which is called a game.

Okay. So to formulate a social problem as

a mathematical model of a game, you have to specify three items.

First, you have to specify who participates in the social problem.

So you specify, who are the players?

Players.

So, if any individuals participate in this social problem we call them players.

Player number 1, player number 2, up to player number N.

Okay, so first you specify who participates in the social problem.

And secondly you have to specify what is possible.

What each player can do in a social problem.

That is called strategy.

Okay?

So each player in a social problem takes some action or strategy.

So strategy of player i.

Is denoted by a sub i, okay?

And do you have to specify the range of possible strategies for each players.

So we need to specify what is called the strategy set.

Okay, so it's large A, sub i.

Okay, this is a set.

A sub i is a set which contains all possible strategies of player i.

So in item number 2,

you specify what kind of actions can possibly be taken by each player.

So that's the second item.

Thirdly, you specify the payoff to each player.

Okay so let's denote each player's payoff by, g sub i.

And your payoff, your profit or

your benefit, usually depends on what everybody does, okay?

So, it may depend on player number 1's action.

It may depend on player number 2's action, up to action or

strategy chosen by the last player, okay?

So this player function represents, the nature of a strategic interaction.

What is best for you depends on, usually depends on what others do.

Well, any social problem can be formulated as a simple mathematical model,

which specifies players strategies and payoffs.

This is what I call a game in game theory.

Well in the previous lecture I posed two questions.

Question one was about political struggle of two parties and

question number two was about the traffic.

So let's try to apply our idea.

Let's try to formulate those social problems as a game.

So let's look at the first question.

Political struggle between the Democrats and the Republican, okay?

So, obviously, there are two players.

Player number 1 is a Democrat.

And, player number 2 is the Republican party, okay?

That's fine.

And what about the possible set of strategies?

Well, this suggestion is very, very, very complex.

So the republican party can do many many things in political campaign.

And also democrats has many ways of persuading voters to vote for them.

So the scope of, of strategy is not so obvious.

And here we need some simplification, okay?

We have to look at the crucial aspect of political campaign.

And even those critical aspect can be formulated as

a symbol of model of game, okay?

So the item number two, the scope of strategies, what is the set of

all feasible strategies, in this particular instance it's not so obvious.

So you need to simplify the reality.

And you have to capture the essence of reality in the simple model of gain, okay?

And let me turn to the third item.

Specify payoff to each player.

In our game, what the payoff is for Democrats and

Republican, it's not so clear.

Okay, so again, you have to look at the essence of reality and

formulate the essential part of their benefit as a pay out function.

Okay, for those very, very complex tasks.

And I'm going to explain how to do that at the end of my lecture in this week.

So let's go to the second question.

Okay, second problem.

The traffic problem.

This is much easier to formulate.

The first use specify who participates in the traffic game.

Okay, so players,

let's say the players are the drivers commuting from city X to city Y.

There may be thousands of players in this traffic game.

That's item number one, to specify who participates in, in the game.

Item number two, strategies of each player.

Well, if you are commuting from city X to Y, you can take many different routes.

It may be this is the way you go from city X to Y, route a.

That is a possible strategy for you.

Route na, route b, you can also take this route b to go from City X to Y.

And there are finitely many routes coming from X to Y, so the set of strategies for

each player is the set of roads, coming from city X to city Y.

Okay, so

item number two strategies is very easy to specify in this particular social problem.

Okay, lastly you have to specify payoffs.

Okay, in this traffic problem each driver tries to

minimize the time to destination, okay?

So you can say that the payoff to each player,

is minus negative of traveling time.

If you maximize your payoff, you are minimizing the traveling time or

commuting time to destination.

So this traffic problem is, clearly formulated as a game

which specifies players, Strategies and payoffs.

Okay, so, after, after formulating a social situation as

a mathematical model of game, then you have to solve.

You have to find a solution to the game,

you have to predict how people will behave.

So then, game theory takes a unified approach.

All of those associated problems have the prolonging two features.

Number one, individuals try to do their best against others.

Number two, but they try to do their best under a certain set of rules.

So, all those social problems share the common futures.

One and two.

So there might be a general theory which can be

applicable to all those social problems, okay.

In the mathematical theory of, in the mathematical formulation of a game,

game specifies players' strategies and payoffs.

The question boils down to, finding a unified solution to all games, okay.

So are there any general theory which is applicable to all those social problems?

That's the question game theory poses.

This is what I called unified approach.

And this was the question posed by the founder of game theory,

Von Neumann and Oskar Morgenstern, okay.

They post this question.

They've, they argued that all social problems can be formulated as a simple

mathematical model, and maybe we can find unified principle or

governing principle which can be applied to all those social problems.

Okay, so this is a picture of the fathers of game theory, von Neumann,

a computer scientist and mathematician and fill, [INAUDIBLE] of physicist,

and Oskar Morgenstern a professor of economics at Princeton University.

Okay together they published a book called, The Theory of Games and

Economic Behavior, back in 1944.

That was the beginning of game theory.

Welcome to Game Theory

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