Games theory problem

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Anonymous

Games theory problem 2016-03-01 03:43:10 Post No. 16863362

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Games theory problem 2016-03-01 03:43:10 Post No. 16863362

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Consider a game in which there is a prize worth $30. There are three contestants, A, B, and C. Each can buy a ticket worth $15 or $30 or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticket-purchase decisions, the game organizer awards the prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket if there is only one such player or is split equally between two or three if there are ties among the highest-cost ticket buyers.

Am i right or wrong?

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What's the question, you just need to break down the possible results?

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Consider a game in which there is a prize worth $30. There are three contestants, A, B, and C. Each can buy a ticket worth $15 or $30 or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticket-purchase decisions, the game organizer awards the prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket if there is only one such player or is split equally between two or three if there are ties among the highest-cost ticket buyers.

The question is there but I have no ideas to solve it

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1. Show this game in matrix form.

2. What are three playersÃbest responses?

3. Find all pure-strategy Nash equilibria.

Sorry Here are the questions

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>>16863412

That's not a question, that's just an explanation of the game, but if you're just supposed to build a matrix for the game, then ya, that looks fine

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>>16863433

However, After doing this Matrix table.

I can't find the nash equilibria

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>>16863439

You're formulating the question in a way that makes this difficult. Pretend that there are only two players, A and (BC), and go from there

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Also you definitely forgot to take into account that buying a ticket costs money in your payout chart. If the prize is worth $30 and you spend $30 buying it, you're coming out at BEST even. Spending $30 is probably not the best strategy, like ever.

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The question is, what is the best strategy to ensure that your winnings are at least higher than your losses.

Even if you get a $30 and win, you're total gains will be zero. So nobody should buy that.

If you get a $15 ticket, your maximum gains will be from zero to $15.

If you get a $0 ticket, your maximum gains will be from zero to $10.

$15 it is.

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>>16863469

If nobody buys a ticket, the prize isn't awarded, so the most you can win by not playing is $0.

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>>16863445

Do you mean I should it on that way?

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>>16863471

>>16863469

Oh, if you get a $15 ticket, you might get a -$5 loss, if all get the same.

So scratch that answer!

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>>16863473

That looks right.

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>>16863473

The choices made by A, B, and C are independent. Also, the payout matrix for every player is identical. So for simplicity, draw it like this, and without loss of generality only focus on A.

Paying $15 is never worse than paying $30, and in some cases is better, so we say that $15 weakly dominates $30, and thus can rule it out in all cases.

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>>16863495

I made a new matrix table

Is it right or wrong??

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>>16863505

No, because B and C can pick different choices, which you don't have reflected in your graph

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Then i have no ideas how to solve this game..

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>>16863535

Okay the full graph looks like this, but it's just a bit overkill since a bunch of the options are technically the same.

Like the column (0, 15) is pretty much the same as (15, 0) except with the players reversed, so we really only needed the one, but I digress.

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>>16863558

Obviously nobody is ever going to pay $30 since $15 is always either better, or no worse. We would say that 15 "weakly dominates" 30. This means we can remove every strategy involving $30 from the chart, since nobody is ever going to pick it.

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>>16863569

Thanks for your help. It helps me a lot

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>>16863569

So then we underline everybody's best response to the other players strategies. (Note: in the case of ties, both are considered "best responses"). A Nash equilibrium occurs whenever no player can improve by switching.

In the chart, I've underlined all the best responses, showing that Nash equilibria occur at

(0, 0, 15)

(0, 15, 0)

(0, 15, 15)

(15, 0, 0)

(15, 0, 15), and

(15, 15, 0)

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>>16863613

It occurs to me that with how many ties there are and how weird the chart looks, it's maybe not so obvious.

For A, I just underlined its payout if it was the best in the column.

For B, I underlined its payout if it's the better of cells 1 and 3 or 2 and 4 in each row.

For C, I underlined its payout if it's the better of cells 1 and 2 or 3 and 4 in each row.

So it all lines up based on what the better option is, given that the other 2 players stay the same.

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The only winning move is not to play. The odds of losing money are higher than not no matter what.

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>>16863362

What class is this for? It's obviously not for a games theory class, because this is intro shit we did in comp sci.

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>>16863771

I don't know why. It says it is a game theory course by computing Department

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You don't have to draw the stupid table. Everyone would buy 0.

>buy 30

>win 30

>net 0

>fucking why

>buy 15

>everyone buys 15

>win 30/3=10

>have lost 5

>buy 0

>net 0

However, that's if using pure logic. In behavioral science, the risk-takers would buy the 15 ticket and the risk-adverse would buy 0, and it just depends on the group makeup if the risk takers win anything.

I love game theory, but it has a time and a place, and that place is not when the answer is so simple you can arrive at it in 20 seconds in your head.

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