Pure Strategy SPNE

Question

Quick question here:

I am evaluating an extensive-form game with five subgames. Four of the five subgames are easily handled. However, the fifth subgame is making me second guess myself.

My question:

Can there be a pure strategy subgame perfect N.E. in an extensive form game if there is a subgame where a player is indifferent between actions?

So, this fifth subgame is a node followed only by payoffs. The player deciding between playing L and R will generate an equivalent payoff whether he plays L or R. So, he is indifferent. Since this player is indifferent at this node, it makes me think I cannot define a SPNE in pure strategies.

Also, I know that a SPNE in pure strategies must be a backward induction solution. Well, since this node is considered first during backward induction, it seems to nix a backward induction solution to the game.

Is this correct? Can anyone clarify?

Answer 1

Every finite full information game has a subgame perfect equilibrium.

A strategy profile is not an equilibrium if at least one of the players has a 'better move'. If they only have one that is exactly as good as their current move they have no incentive to change.

About backward induction:
It is not nixed in such cases. If a player is indifferent between two choices you mark both branches as both choices are subgame perfect. You can also see from this that by following subgame perfect strategies you will reach from the root to the end so a SPNE exists.

Indifference between some actions on the part of a player is a necessary but not sufficient condition of having multiple SPNE in a finite full information game.