In a game with imperfect information, can there be no subgame-perfect equilibrium?

Question

In our Game Theory class, we learned that games can have multiple Nash equilibria and multiple subgame-perfect Nash equilibria ($SPNE$). In one of our example problems, however, we came across a game with imperfect information and apparently no $SPNE$s whatsoever.

Our teacher wasn't sure how to explain this, and while we found a site that said games with perfect information always have at least one $SPNE$, we couldn't find anything that said whether or not games with imperfect information always have $SPNE$s as well.

Is it possible for a game with imperfect information to have no $SPNE$s? If not, can you help us determine the $SPNE$ of this particularly vexing problem?

The problem itself is depicted in the picture below.

This is a picture of our progress so far with a possible SPNE $\{DA,Rlx\}$ listed on the right. Apologies for the quality of the picture.

Answer 1

Nash's existence theorem states that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium (NE). (This means if there is no pure strategy NE of such a game then then there must be a mixed strategy NE.)

It follows that there must be a SPNE (possibly involving some randomization) for your game. This is because any subgame of your game has a finite number of strategies and so has a Nash equilibrium (and an SPNE is defined as a strategy profile where players are playing a NE in every subgame).

If there is no SPNE in pure strategies (I haven't checked), then there must be one involving mixing. It turns out there are actually three SPNE, one of which is in pure strategies.

I will label the actions of player 2 in the bottom right subgame as $X$ and $Y$.

• The subgame in the bottom left after history $(U,L)$: Unique SPNE strategies $(A,(l,y))$ with outcome $(A,L)$ payoffs $(2,1)$

• The subgame in the bottom right after history $(D,R)$: Three Nash equilibria. Two pure strategy NE with strategy profiles $(a,X)$ and $(b,Y)$ with corresponding payoffs $(2,1)$ and $(1,2)$. One mixed strategy NE where player $1$ plays $a$ with probability $2/3$ and player $2$ plays $X$ with probability $1/3$, and expected payoffs are $(2/3,2/3)$.

Thus we are left with three possible payoffs when player $1$ plays $D$ and player $2$ plays $R$.

Suppose they play $(a,X)$ with payoffs $(2,1)$ after history $(D,R)$. Then payoffs are:

          2
       L           R
  U   2,  1      3/2, 0
1
  D   3/2, 0     2, 1

This has two pure strategy NE. They are (U,L) and (D,R). It also has a mixed strategy NE where each plays each of the actions with probability $1/2$. The expected payoffs in this SPNE are $(7/4,1/2)$.

Now suppose they play $(b,Y)$ with payoffs $(1,2)$ after history $(D,R)$. Then payoffs are:

          2
       L           R
  U   2,  1      3/2, 0
1
  D   3/2, 0     1, 2

Here there is a pure strategy NE: $(U,L)$. The payoffs in the corresponding SPNE are $(2,1)$.

Finally, if they play the mixed strategy NE with expected payoffs $(2/3,2/3)$ after history $(D,R)$ we get the game:

          2
       L           R
  U   2,  1      3/2, 0
1
  D   3/2, 0     2/3, 2/3

The Nash equilibrium of this game is (player $1$ has dominant strategy $U$) $(U,L)$. The corresponding SPNE has payoffs $(2,1)$.

Note that I have not written down the SPNE strategies in each case (they are a bit messy), but just look back at the previous steps of working to see what they are. For example, in the second case the SPNE strategy profile is $((U,A,b), (L,l,y,Y))$