# SPNE and Pareto Optimality

• "The SPNE of a sequential game might not necessarily be Pareto Optimal"

I understand the definitons of Nash Equilibria and Pareto Optimality and that these are not synonymous concepts. An example in reference to this question would be a Firm-Union Wage Setting Game, in which there might be $$w$$ offers which provide greater payoffs to both the firm and union. Please share some other examples for the same. Thanks!

• Perhaps this might help: In layman terms, a Pareto optimal point is the solution of an optimization problem. On the other hand, Nash equilibrium is a fixed point. Apr 7, 2019 at 17:09

## 1 Answer

Proof of "the SPNE of a sequential game might not necessarily be Pareto Optimal"? I don't get it, your example is a proof of this statement. So what else do you need?

If you need another example, just take the prisoner's dilemma, and turn into a sequential game with imperfect information. Then, the NE is equal to the SPNE and you have an equilibrium without Pareto optimality

• I understand. Providing one counterexample would be enough for the statement. Perhaps I should edit my question. Thank you for this example. It helps. Apr 7, 2019 at 17:20
• Sorry if i sounded a little rude. But now you get it, right? Your "theorem" is "Not all SPNE are Pareto Optimal". So, if there exists one SPNE that isn't Pareto Optimal, your "theorem" is proved. Apr 7, 2019 at 17:23
• No worries! I was a bit hasty whilst posting and forgot that a counter example would suffice as a formal proof. Again, thank you for the Prisoner's Dilemma example. Apr 7, 2019 at 17:25