<p>Looking for <em>the</em> subgame perfect equilibrium here is hopeless; there really is more than one (in spite of what the assignment says).</p>
<p>When you use backward-induction, you have to use both. There is a subgame perfect equilibrium in which player <span class="math-container">$2$</span> chooses <span class="math-container">$d$</span> at the correspond decision node and a subgame perfect equilibrium in which player <span class="math-container">$2$</span> chooses <span class="math-container">$e$</span> at the same decision node. As a consequence, player <span class="math-container">$1$</span> makes different choices in their first decision node in these two subgame perfect equilibria.</p>
<p>By the way: An equilibrium specifies a strategy for each player, and a (pure) strategy specifies a decision at each information set. For example, player <span class="math-container">$1$</span> has four strategies:  <span class="math-container">$LW$</span>, <span class="math-container">$LE$</span>, <span class="math-container">$RW$</span>, <span class="math-container">$RE$</span>. If player <span class="math-container">$1$</span> moves left initially, their choice at the other decision node will have no effect on the resulting play. However, it is still important for determining whether a strategy profile is subgame perfect or not.</p>

CC BY-SA 4.0
2021-03-30T22:46:07.600Z
2021-03-30T22:46:07.600Z
Michael Greinecker
<p>For a homework assignment, I need to find the subgame perfect equilibrium. The assignment asserts that there is only one subgame perfect equilibrium in this problem, but I am stuck between two possible options.</p>
<p><a href="https://i.sstatic.net/BtokU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BtokU.png" alt="enter image description here" /></a></p>
<p>I found three Nash equilibria at (2,1), (3,2), and (1,2). The output of the left side is (2,1), but I am unsure if the output of the right side is definitively (3,2) or (1,2). Through backwards induction, P1 would choose (1,2) over (0,3), but I don't know if P2 would choose (3,2) or (1,2) since it's an equally good output for P2.</p>

<game-theory><nash-equilibrium>
Can't find the SPNE
John Laster
# Can't find the SPNE

(Asked by: username_0 on 2021-03-30)

For a homework assignment, I need to find the subgame perfect equilibrium. The assignment asserts that there is only one subgame perfect equilibrium in this problem, but I am stuck between two possible options.

[![enter image description here](https://i.sstatic.net/BtokU.png)](https://i.sstatic.net/BtokU.png)

I found three Nash equilibria at (2,1), (3,2), and (1,2). The output of the left side is (2,1), but I am unsure if the output of the right side is definitively (3,2) or (1,2). Through backwards induction, P1 would choose (1,2) over (0,3), but I don't know if P2 would choose (3,2) or (1,2) since it's an equally good output for P2.

**Tags:** game-theory, nash-equilibrium

---

## 1 answers

### Answer by username_1 on 2021-03-30. Score: 3 | Accepted answer

Looking for *the* subgame perfect equilibrium here is hopeless; there really is more than one (in spite of what the assignment says).

When you use backward-induction, you have to use both. There is a subgame perfect equilibrium in which player $2$ chooses $d$ at the correspond decision node and a subgame perfect equilibrium in which player $2$ chooses $e$ at the same decision node. As a consequence, player $1$ makes different choices in their first decision node in these two subgame perfect equilibria.

By the way: An equilibrium specifies a strategy for each player, and a (pure) strategy specifies a decision at each information set. For example, player $1$ has four strategies: $LW$, $LE$, $RW$, $RE$. If player $1$ moves left initially, their choice at the other decision node will have no effect on the resulting play. However, it is still important for determining whether a strategy profile is subgame perfect or not.


---

