3-3 “Payoffs” in a Game: What Exactly Are Those Numbers?

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Do curso por The University of Tokyo

Introdução à Teoria dos Jogos

736 classificações

The University of Tokyo

Introdução à Teoria dos Jogos

736 classificações

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.

Na lição

Rationality, Knowledge, and Evolution in Games

In Module 3, we will dig deeper into the relationship between rationality and Nash equilibrium. We will consider the whole spectrum of possible intellectual capacities of players, spanning the range from unlimited ability for sophisticated reasoning to absolute zero intelligence. In the end, you will see that Nash equilibrium can emerge under a fairly wide range of intellectual capacities of players.

3-1 Digression: The Card Game Revisited 11:29

3-2 Digression: How You Played the Card Game and Addressing the Concerns about Game Theory15:57

3-3 “Payoffs” in a Game: What Exactly Are Those Numbers? 9:31

Conheça os instrutores

Michihiro Kandori
Professor
Faculty of Economics

So now, I'm ready to examine the relationship between

player's intelligence and and their behavior.

Okay?

And in games where we play as trying to maximize their payoff and the payoffs

as often given by a payoff table and you see those numbers, three, two and zero.

And I'm going to explain exactly what those numbers mean.

That's the starting point in examining the relationship between

intelligence of players and the outcome.

Okay. So

what are the payoffs in a game, those numbers three, two and zero.

Well, they may be prize or profit.

So $3, $2, $0.

That's one possibility.

Or it may be present in benefits in some sense of the player or

the level of satisfaction.

Okay?

So the, the concept of payoff is not so clear.

What those numbers replacing is not so clear?

So let me give you precise explanation what those numbers are or

what those number should be.

Okay.

So there are two cases.

One is easier case, the other is a harder case.

The easier case concerns the situation where there's no uncertainty.

Okay. So

let me explain what I mean by a situation without any uncertainty.

So the situation I'm thinking about now has no random event.

So nothing like weather or stock price is present in the game.

So no random events in the game and no random behavior in the game.

So unlike the game of rock,

paper, scissors people are not taking random strategy.

Okay?

So when there is no uncertainty, it's very easy to specify player's payoff.

In this situation, without any uncertainty to determine a player's payoff,

it's very easy.

You can just assign larger numbers for better outcomes.

The better the outcome, the larger the number.

That's your payoff.

So let me explain what I mean by a very simple example.

Let's suppose the game has three outcomes.

Best outcome, worst outcome and worst outcome for you.

Okay?

And you have, your task is to assign payoff to this outcome,

payoff to the middle outcome and payoff to the worst outcome.

Well, you can assign larger number for better outcome.

So one way is three for best, two for the middle and one for the worst.

Okay?

Larger numbers for better outcome, so this is one possibility.

But there are lots of other possible ways of assigning payoffs.

Well, this is another way.

100, 8 and minus 1.

And still this is another way, 20, 19 and 0.

Okay?

So game theory says, any one of those choices is fine.

Okay?

So, and let me explain why.

So, any assignment of numbers is fine,

as long as you assign larger numbers for better outcomes.

Why?

So no matter which presentation or which choice you choose, there let's suppose 20,

19 and 0 that is one way to assign larger numbers for a better outcome.

Okay.

So maximizing your payoff means choosing better behavior.

Okay.

It really doesn't depend on whether this number is 20 or 100.

As long as this number here is larger than this number and

as long as this number is larger than this number,

payoff maximization means choosing better outcome.

Okay?

Okay.

So to determine players payoff in a situation without any uncertainty,

you can just assign larger numbers to better outcomes for the player.

Because that is how you should specify payoffs in

a game when there's no uncertainty.

Okay.

Suppose you have some uncertainty in a game,

then you should assign payoffs more carefully.

So let me explain how I do that.

Okay.

So let's consider situation with uncertainty.

So maybe there are some random events like weather or stock price in the game or

maybe people are taking random behavior like rock, paper and scissors.

So, in that kind of situation player may like risk.

A player's may dislike risk and

you have to somehow represent those two differential situations.

Okay.

So let's suppose you have a randomness, random events in, in a game and

suppose a player is facing a choice between a lottery and some other outcome.

So, if a player chooses a lottery with probability one-half, he gets $10.

And with the other probability one-half, he gets zero.

So uncertain event, the outcome of the lottery is here.

So this is one choice, choosing this lottery.

Another choice, the other choice is to get $5 for sure.

Okay.

So let's examine what is best for you and there's no right answer.

It really depends on how you like or dislike the risk.

Okay.

So, if you choose this lottery here on average, you can get $5.

Okay? So half the time, you get $10.

Half the time, you get zero.

On average, you get $5.

This is the expected payment of this lottery.

On the other hand, if you take the other choice you get $5 for sure.

So on average, those two choices give you exactly the same amount.

What's the difference?

Well, difference is obvious.

A lottery has some risk and this choice here doesn't have any risk.

It's a safe choice.

So if you like risk, if you are a,

a gambler it's more fun to play this lottery.

So you may choose risky choice.

Okay. So

risk-loving people, gamblers may choose this lottery.

But if you are conservative or

cautious, you may like the other choice, the safe choice.

So risk-averse players choose the the second choice.

Okay.

So we have to represent those two different behaviors somehow and

here is a very easy way.

Okay?

First, you assume that players get some utility from outcome.

Okay.

So if you choose this lottery, outcome will be either $10 or $0.

So this, you get some utility out of $10.

Okay?

And if outcome was $0, you get some utility of $0.

Okay.

So with equal probability, you either get this utility or that utility.

And if you choose the other alternative,

you'll get this utility coming from $5 for sure.

Okay?

And then after assigning those numbers, utilities,

you should assume that players maximizing expected utility.

Okay? So, if player chooses this lottery half of

the time, he gets utility from $10 dollars in lieu of ten.

And half of the time, he gets the utility coming from $.

So on average, his satisfaction or expected utility is this one.

On the other hand, if a player chooses the other alternative, he gets $5 for sure.

So he gets satisfaction or utility from $5 for sure, okay?

So if number here is smaller, then the average of satisfaction ten and zero.

So if $5 for sure have small utility, then you choose lottery,

because lottery has large expected utility.

Okay?

And if your utility from $5 for sure is larger compared to the average

of utility of ten and zero and a then this is better for you.

Okay.

And you choose $5 for sure.

Okay. So let me sum up.

Okay?

So when uncertainty is presented in game situation,

a player's payoff is equal to utility coming from the outcome and

you assume that players are maximizing expected utility.

And by specify, specifying those numbers, utility is coming from outcomes,

you can describe if a player likes or dislike the risk.

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