Prisoner’s dilemma

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An exercise in calculating economic outcomes by means of metaphor, the prisoners dilemma was developed at the RAND corporation in the 1950s by those splendid brainboxes as a way of predicting individuals’ behaviour in situations requiring trust among strangers – for very good example, when unacquainted participants buy or sell in an unregulated market. This field developed into game theory.

The original dilemma

Prisoner’s dilemma
payoff table

A cooperates

A defects

B cooperates

A gets 1 year
B gets 1 year

A goes free
B gets 3 years

B defects

A gets 3 years
B goes free

A gets 2 years
B gets 2 years

Two people are charged with a conspiracy[1]. Each is held separately. They cannot communicate. There is enough evidence to convict both on a lesser charge, but not the main charge. Each prisoner is separately offered the same plea bargain. The offer is:

  • If A informs B but B refuses to inform on A:
    • A will not be prosecuted at all and will go free
    • B will be convicted of the main charge and will get 3 years in prison.
  • If A informs B and B informs on A:
    • A will get 2 years in prison
    • B will get 2 years in prison
  • If A refuses to inform on B and B refuses to inform on A:
    • A will get 1 year in prison (on the lesser charge).
    • B will get 1 year in prison (on the lesser charge).

The eBayer’s dilemma

eBayer’s dilemma
payoff table’

Buyer cooperates

Buyer defects

Seller cooperates

Buyer gets £50
Seller gets £50

Buyer gets £150
Seller gets -£50

Seller defects

Buyer gets -£100
Seller gets £200

Buyer gets £0
Seller gets £0

The consequence is easier to see in more familiar context. This version Douglas Hofstadter once described as the “briefcase came”, but Millennials will recognise it as the eBayer’s dilemma.

Strangers consider a settling a transaction they have drunkenly agreed on eBay to purchase a set of complete boxed set of the final season of Some Mothers Do ’Ave ’Em on VHS, for £100[2]. The seller must pay and the buyer must post the videos simultaneously. Assume the seller would have sold the videos at any price over £50 (so the transaction represents a £50 profit for her), and privately, the buyer would have been prepared to pay £150 had that been the asking price (the transaction therefore representing a £50 profit for him).

When settling the transaction, each has the following risks:

  • If the seller posts and the buyer pays:
    • The seller will get £100 but will lose a video it values a £50 (+£50).
    • the seller will get a video it values at £150 but must pay £100 (+50)
  • If the seller posts, but the buyer does not pay:
    • The buyer will have video it values at £150 for free (+£150)
    • The seller has given away a video it values a £50 for nothing (-£50).
  • If the buyer pays but the seller does not post:
    • The buyer will have paid £100 for nothing (-£100)
    • The seller will receive £150 and will still have the video it values a £50 (+£200).
  • If the seller does not post and the buyer does not pay:
    • The seller is in its original position (£0).
    • The buyer is in its original position (£0).

Single round

If you play this game in isolation — with someone you don’t know and whom you do not expect to meet again, the payoff is grim: those who cooperate will get reamed. Cooperation is a bad strategy. Your best interest is in defecting on the other guy, because his best interest is defecting on you.

This looks like a bad outcome for commerce. If the rational disposition is to weasel on a deal, how can we have any faith in the market? How, come to think of it, has any kind of market ever got off the ground? Why would anyone take on a sure fire losing bet?

Because trust, faith and confidence changes everything. The single round prisoner’s dilemma stipulates there is no consequence on a bad actor for reneging. The defector is guaranteed to get away with it: these are the rules.

But in real life, one-off interactions with strangers — counterparts whom you are guaranteed never to see again — are rare. Business is the process of cultivating relationships. Establishing trust.

The game theorists found an easy way to replicate that concept of trust: run the same game again. Repeatedly. An indefinite amount of times.

“Iterated” prisoner’s dilemma

The same actors get to observe how each other act, and respond accordingly. If your counterpart defects, you have a means of retaliating: by defecting on the next game, or by refusing to play the game any more with that counterparty.

Now, as well as the short-term payoff, there is a longer-term payoff, and it dwarfs the short term payoff. If I defect once, I earn £150. If I cooperate a thousand times, I earn £50,000. If I defect first time round, sure: I am £100 up, but at what cost: if my counterparty refuses to play with me again — and if she tells other players in the market — I will struggle to make much money. No one will trust me.

See also

References

  1. Whether or not they are guilty is beside the point. If it helps you empathise with their predicament, assume they’re innocent
  2. The eBayers don’t have to be drunk, but as we all know, most people on eBay are, so this is a bit of dramatic colour.