Game Theory 101 (#14): Infinitely Many Equilibria

Sometimes, a game can have infinitely many equilibira. This lecture provides an example and also illustrates the concept of partially mixed strategy Nash equilibria.

[Gametheory]

I talk more about the meaning of the payoffs in lesson 1.1 of Game Theory 101: The Complete Textbook. You should check that section out. (It's in the free sample Amazon provides, so you should do this regardless of whether you want to buy the book.)

[PunmasterSTP]

Infinitely many? More like “Interesting like crazy!” These are some of the coolest videos I’ve ever seen on YouTube; keep up the amazing work!

[Gametheory]

No. Whatever values the players have regarding the other players' outcomes is already reflected in their payoffs. So if it says player 1 receives 3 if he goes up and 3 if he goes down, then he is completely indifferent between those outcomes.

[GeneralSeptem]

I've noticed something interesting about this game. If player 2 employs *any* mixed strategy, then player 1 is no longer indifferent and will always choose "up". Save for the small percent of the time when she picks right, wouldn't player 2 always prefer this outcome? (I do get that this is not an equilibrium, as for whatever p_right she chooses, she'd do better with a smaller p_right, except when p_right -> 0 and player 1 becomes indifferent)

[feameldo]

What a cute game. Looks like an exotic form of Ultimatum with a catch: a pure Ultimatum can be played "fair", but what is "fairness" here? If player2 "bribes" player 1 into playing Up, by playing Right 1/12 of time (1 = player2's win from Up-Left, 12 = player1's win from Up-Right) - will it be fair? Will 1/9 (9 = 12 - 3, difference between player1's wins) be fair? Will 1/2 be fair? Will the players agree on that? ...it's a cute and thought-provoking game, I'll grant it that.

[guedes_]

On that last video, we did not have an infinitely many equilibria, because when we eliminate strategy right of the game, player 1 will always prefer to play up.

[Rayswai]

Theoretically, I understand that "right" is strictly dominated. But the way I see it, it could be preferable for P2 to play right since P1 then would play UP, and P2 gets 0. 0 is better for P2 than randomly either 1 or -2. What is it that i don't understand?

[johnInception]

Is there some way that player two could incentivise player one to choose "up" by occasionally choosing "right"?

[vachikkhachatryan]

Is there a infinitely many strateges in zero sum games?

[sivaprasath]

Is there any real time example ...where this strategy is used??

[mauritsbol]

Btw. There is only one true equilibrium here. Red should play right 1% of the time. That way, she forces player one to play up all the time. 99% of the time, she cashes in.

[housesarentonsale]

Player 1 doesn't care what happens to player 2.

[housesarentonsale]

To be honest they are not opponents really opponents at all. Otherwise Player 1 would have gotten more points on that square.... Maybe he already has.

[dmit]

Is it true that Ballet-Fight game from #10 also has an infinite number of [mixed] equilibria?

[xinyizhang]

If it doesn't matter with which player I begin the analysis, wouldn't down be weakly dominated by up since 3=3 and 12>2, so that player 1 would never play down? And if down is eliminated, it only makes sense for player 2 to play left instead of right since 1>0, then there is only 1 equilibrium and not indefinite right?

[yaarbach]

do you go over cournot & stackelberg models?
How about downs & hotelling?

[HuffyTheMan]

Wouldn't Player 1 always want to play down because it would leave their opponent worse off?

[treeman]

What if player 2 wanted to punish player 1 for playing down rather than up? Then Player 2 might play right in response to player 1's down