<p>You could represent the game in simultaneous form (matrix) to see all possible equilibria. If there is only one, then that's your subgame-perfect Nash equilibrium (since there are no other equilibria). If there are multiple equilibrias in the simultaneous representation, the extensive form allows you to see if some paths would never be reached given the structure of the game.</p>

<p>Essentially, the extensive form "proves" which paths would never be reached through ruling them out. The one path that is reached is your SPNE. An SPNE has a more strict definition. There is really no shortcut around the arithmetic in the extensive form since the definition of SPNE forces you to check for the one viable path.</p>

CC BY-SA 3.0
2017-04-01T17:28:19.567Z
2017-04-01T17:28:19.567Z
rrod18
<p>If I have an specific extensive game, with only a finite set of strategies, how can I demonstrate that the game always have a subgame-perfect equilibrium in pure strategies? My first intuition was to show that in every subgame a pure strategy it's a solution to the problem (maybe not unique) and then there is always a pure strategies equilibrium. But I'm not sure. </p>

CC BY-SA 3.0
2017-03-29T22:23:24.883Z
I'm not sure the statement is true in general. The only SPE in matching pennies is in mixed strategies.
Herr K.
CC BY-SA 3.0
2017-03-30T02:13:04.307Z
Perhaps the OP has in mind finite extensive form games with perfect information, for which I believe the claim is true.
Theoretical Economist
CC BY-SA 3.0
2017-03-30T15:05:08.000Z
This is true in mixed strategies, I think.
123
CC BY-SA 3.0
2017-03-31T15:07:52.623Z
I'm not saying that it's true for all games, sorry if I did not explain myself good. Its if I had an specific extensive game.
hllspwn
CC BY-SA 3.0
2017-03-31T15:39:19.407Z
Then you should tell us what this specific game is, and show that you've made some effort on trying to prove the result.
Herr K.
CC BY-SA 3.0
2017-03-31T17:56:04.480Z
@123: Only in the sense that pure strategies are special cases of mixed strategies. For example, the only SPE in Prisoners' Dilemma is in pure strategies.
Herr K.
<game-theory><nash-equilibrium><extensive-games>
How to demonstrate that a game always have a subgame-perfect equilibrium in pure strategies?
hllspwn
# How to demonstrate that a game always have a subgame-perfect equilibrium in pure strategies?

(Asked by: username_0 on 2017-03-29)

If I have an specific extensive game, with only a finite set of strategies, how can I demonstrate that the game always have a subgame-perfect equilibrium in pure strategies? My first intuition was to show that in every subgame a pure strategy it's a solution to the problem (maybe not unique) and then there is always a pure strategies equilibrium. But I'm not sure.

**Tags:** game-theory, nash-equilibrium, extensive-games
> **Comments:**
> **username_1**: I'm not sure the statement is true in general. The only SPE in matching pennies is in mixed strategies.  
> **username_2**: Perhaps the OP has in mind finite extensive form games with perfect information, for which I believe the claim is true.  
> **username_3**: This is true in mixed strategies, I think.  
> **username_0**: I'm not saying that it's true for all games, sorry if I did not explain myself good. Its if I had an specific extensive game.  
> **username_1**: Then you should tell us what this specific game is, and show that you've made some effort on trying to prove the result.  
> **username_1**: @123: Only in the sense that pure strategies are special cases of mixed strategies. For example, the only SPE in Prisoners' Dilemma is in pure strategies.  

---

## 1 answers

### Answer by username_4 on 2017-04-01. Score: -1

You could represent the game in simultaneous form (matrix) to see all possible equilibria. If there is only one, then that's your subgame-perfect Nash equilibrium (since there are no other equilibria). If there are multiple equilibrias in the simultaneous representation, the extensive form allows you to see if some paths would never be reached given the structure of the game.

Essentially, the extensive form "proves" which paths would never be reached through ruling them out. The one path that is reached is your SPNE. An SPNE has a more strict definition. There is really no shortcut around the arithmetic in the extensive form since the definition of SPNE forces you to check for the one viable path.


---

