Math Is Fun Forum

Karimazer1

Guest

Urgent Game Theory Help....

I am at the end of the 'simple' game theory stuff. After doing sequential/extensive and normal form, this pops up:

(The following link is the Normal form representation of the game..) some of them aren't clear, please check with my below writing!
tinypic.com/r/25qduhl/5

Just so you can check incase you write down, for the game on the LEFT, the payoffs are meant to write (since its done in paint)...

1,1 10,0
0,10 8,8

and for the second game

4,4 10,200
200,12 10,400

The question is....'player 2 moves first and chooses ONE of the two games to play with player 1, find the sub game perfect nash equilibrium of the 2 stage game'.

I can do this...I worked backwards....but it's a confusing answer (as I think there are two....)

Please try it and see if you understand what I mean. I really want to move on! Thank you

Karimazer1

Guest

Re: Urgent Game Theory Help....

If you need to me to clarify the question/ the 2 normal form games being chosen between let me know

Karimazer1

Guest

Re: Urgent Game Theory Help....

no one can put their game theory hats on for me? =d

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Urgent Game Theory Help....

hi Karimazer1

Welcome to the forum! 

I don't have a game theory hat but, to avoid 'deathly silence' I looked it up at

http://en.wikipedia.org/wiki/Nash_equilibrium

So I'm on a steep learning curve here. 

I'll say what I think; you can explain to me why I'm wrong and maybe that will make it clearer for us both. 

I put your matrices into Excel and did a screen shot.  See below.  I've added numbers to make it clear whose move is whose. 

So player 2 chooses game 2 because the winnings are so much better.

'He' then chooses strategy R as the winnings are so much better.

Player 1 has no reason to choose either T or M as 'he' cannot gain more with either.

Let's say 'he' chooses T.  Player 2 cannot improve his win.  Player 1 cannot improve his win.  Therefore R,T is in Nash equilibrium.

Let's say he chooses M.  Player 2 cannot improve his win.  Player 1 cannot improve his win.  Therefore R,M is in Nash equilibrium.

Let's just try the alternative strategy for player 2.

Player 2 chooses L.

Player 1 will choose M.  Player 2 can improve his win by switching to R.  Player 1 cannot improve his win.  Therefore L anything is not in Nash equilibrium.

Is that a correct interpretation?

Should I analyse the 'player 2 chooses game 1' possibilities as well?

Please post back.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Karimazer1

Guest

Re: Urgent Game Theory Help....

Bob,

That is absolutely correct - well done. Except there are 3 Nash Equilibrium.

(The only reason I am personally confused, is because we are supposed to give one answer, so I'm just unsure if we choose either T or B)

Your logic and reasoning is spot on.

For your last question; yes, you need to look at the player 2's game 1 possibilities because this shows that there is another NE and that's what you eliminate by your first step in the general case. What you did was just use your intuition, which was correct for the Subgame NE answer, not if you were finding all NE.

If player 2 chooses game 1, and chooses L...Player 1 will choose T. (Underline the payoff, in this case 1 in the top left cell)
If player 2 chooses game 1, and chooses R...Player 1 will choose T. (Underline the payoff, in this case 10 in the top right cell of game one)
If player 2 chooses game 2, and chooses L...Player 1 will choose M. ( Underline the payoff, in this case 200 in the bottom left cell of game two)
If player 2 chooses game 2, and choose R...Player 1 will choose M&T (indifferent, they're equal as you noted) (Underline 10 twice, the one in the top right and top left of game 2)

Now that's what you compare...(to get what you got by using your brain, I.e. that game two is always better for player 2)...
Player two knows that if chooses GAME 1, he knows he can only get 1 or 0...(1 if he chooses Left, 0 if he chooses right)...but in game 2 he can get 10 or 200 or 400....which are always better. So he chooses game 2 and he plays R. Again, I'm confused because this question requires one answer...so I'm not sure if I just choose either T or M for player 1.. or take an average (i.e. the expected payoff).

Anyway, that's for SUB GAME Perfect Eqm. (ALL SGNE are NE but not all NE are SGNE...this is key)

For Nash equilibrium, go back to your tables... you shoud have 5 numbers underlined. Now do the same for player 2...I.e.,

If player 1 chooses T (game one), player 2 chooses L (as 1>0)
If lpayer 1 chooses M (game one), player 2 chooses L(10>8)
If player 1 chooses T (game two), player 2 chooses R (200>10)
If player 1 chooses M(game two), player 2 chooses R(400>10)

Underline these 4 payoff for player 2.

There are three outcomes, TL (game 1), TR (game 2) and MR (game 2). These three are nash equilibrium. Why?
What we've done above...is find each players' BEST RESPONSE FUNCTION to the OTHER PLAYERS CHOICE.

One way of defining NE is that it is a payoff where EACH PLAYERS BEST RESPONSE IS THE BEST RESPONSE OF THE OTHER PLAYER.

Therefore there are 3 NE in this game.

I hope that clears it up, you obviously understand it..I just thought I'd explain the long method. Obviously, after doing 1/2 of these, you can just skim across rows and columns and underline without writing those above sentences...to get a NE much faster.

I still don't know how to get a sole SGNE for this problem though which is frustrating. Maybe I'll move to another chapter of the book and come back to it.