Given the game like Prisoner's dilemma (infinitely undiscounted) and each person has 3 actions in every period: confess defect and kill, and the payoff matrix have 2 NE as (D,D),(K,K).

Can I still use the property that if in any stage NE is played, then that kind of strategy is SPNE.

It seems work for me to prove that any sequence like dd,kk,kk,kk,dd....randomly combination of dd,kk, is SPNE.

Is that correct? Can I generalize that property?

  • $\begingroup$ How do you ensure convergence of payoff if the stage game is infinitely undiscounted? $\endgroup$
    – Herr K.
    Commented May 26, 2018 at 15:51
  • $\begingroup$ Thanks for your asking. It's my bad that not state clear. I am assuming facing discounting factor as 1 and here we use average payoff instead of discounted future value. comparing the average payoff solving that the payoff is not converge $\endgroup$
    – Chuang Li
    Commented May 26, 2018 at 18:24

1 Answer 1


Though a bit unconventional to assume an infinitely repeated game without discounting, the short answer to your question is: yes. Any sequence of stage game Nash Equilibria is supportable as a SPNE. See, for example slide 11 of this deck, or these notes from Levin, starting on page 6. This is more directly addressed by the definition of SPNEs, found on page 137, Definition 132 of these notes. Or where those Columbia notes suggest on page 142: "For example if there are multiple Nash equilibria in any of the final subgames then there are multiple sub game perfect Nash equilibria in the game itself"

Though you should note that, given your problem's construction (with no discounting), the set of SPNEs grows substantially, as you can see.

  • 1
    $\begingroup$ It is not that unconventional to assume no discounting. $\endgroup$
    – Giskard
    Commented May 27, 2018 at 7:33
  • $\begingroup$ I also edited one of your links. If you put "#page=145" after the URL of a pdf the link will just jump to that page. $\endgroup$
    – Giskard
    Commented May 27, 2018 at 7:36
  • $\begingroup$ Thank you so much for both of your help. That helps me a lot. Not only like three action PD game, but also like the hawk-dove game, any game with multiple NE. I am able to say that as long as stage nash is being played on the equilibrium path, then those kinds of strategies are SPNE. $\endgroup$
    – Chuang Li
    Commented May 27, 2018 at 17:48

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