• Not technically a Prisoners' Dilemma!

  • Game affected by Joker's threat to blow both of them up at midnight if nobody acts

• Both players have a weakly-dominant strategy to Detonate

• What is/are the Nash equilibrium/equilibria?

• We'll stick with these specific payoffs for this lesson

• How can we sustain cooperation in Prisoners' Dilemma?

• For any game with a unique PSNE in a one-shot game, as long as there is a known, finite end, Nash equilibrium is the same

• In experimental settings, we tend to see people cooperate in early rounds, but close to the final round (if not the actual final round), defect on each other

• There are two nearly identical interpretations of infinitely repeated games:
  1. Players play forever, but discount (payoffs in) the future by a constant factor
  2. Each round the game might end with some constant probability

• Since we are dealing with payoffs in the future, we have to consider players' time preferences

• Easiest to consider with monetary payoffs and the time value of money that underlies finance

$$PV=\frac{FV}{(1+r)^t}$$

$$FV = PV(1+r)^t$$

• Example: what is the present value of getting $1,000 one year from now at 5% interest?

$$\begin{align*} PV &= \frac{FV}{(1+r)^n}\\ PV &= \frac{1000}{(1+0.05)^1}\\ PV &= \frac{1000}{1.05}\\ PV &= \$952.38\\ \end{align*}$$

• Example: what is the future value of $1,000 lent for one year at 5% interest?

$$\begin{align*} FV &= PV(1+r)^n\\ FV &= 1000(1+0.05)^1\\ FV &= 1000(1.05)\\ FV &= \$1050\\ \end{align*}$$

• Suppose a player values $1 now as being equivalent to some amount with interest \(1(1+r)\) one period later

  • i.e. $1 with an r% interest rate over that period

• The “discount factor” is \(\delta=\frac{1}{1+r}\), the ratio that future value must be multiplied to equal present value

$$\$1 \text{ now} = \delta \, \$1 \text{ later}$$

• With these payoffs, the value of both cooperating forever is \(\left(\frac{3}{1-\delta}\right)\)
• Value of both defecting forever is \(\left(\frac{2}{1-\delta}\right)\)

• Alternate interpretation: game continues with some (commonly known among the players) probability \(\theta\) each round

• Assume this probability is independent between rounds (i.e. one round continuing has no influence on the probability of the next round continuing, etc)

• If you are playing the Grim Trigger strategy, consider your opponent's incentives:
  • If you both Cooperate forever, you receive an infinite payoff stream of 3 per round

$$3+3\delta+3\delta^2+3\delta^3+\cdots+3\delta^{\infty}=\frac{3}{1-\delta}$$

• This strategy is a Nash equilibrium as long there's no incentive to deviate:

• If \(\delta > 0.5\), then player will cooperate and not defect

• \(\delta > 0.5\) is sufficient to sustain cooperation under the grim trigger strategy
  • This is the most extreme strategy with the strongest threat

• Two interpretations of \(\delta > 0.5\) as a sufficient condition for cooperation:

1. \(\delta\) as sufficiently high discount rate
   • Players are patient enough and care about the future (reputation, etc), will not defect
2. \(\delta\) as sufficiently high probability of repeat interaction
   • Players expect to encounter each other again and play future games together

• Consider a "Forgiving Trigger" strategy:
  • On round 1: Cooperate
  • Every future round: so long as the history of play has been (Coop, Coop) in every round, play Cooperate. Otherwise, play Defect for 3 rounds
    • Punishment, but lasts for 3 rounds, then reverts to Cooperation

• Consider the "Tit for Tat" strategy:
  • On round 1: Cooperate
  • Every future round: Play the strategy that the other player played last round
    • Example: if they Cooperated, play Cooperate; if they Defected, play Defect

• Consider the "Tit for 2 Tats" strategy:
  • On round 1: Cooperate
  • Every future round: Cooperate, unless the other player has played Defect twice, then play Defect