What counts as defection?

by Alex Turner4 min read12th Jul 20204 comments

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Game TheoryCoordination / CooperationRationalityAI
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Thanks to Michael Dennis for proposing the formal definition; to Andrew Critch for pointing me in this direction; to Abram Demski for proposing non-negative weighting; and to Alex Appel, Scott Emmons, Evan Hubinger, philh, Rohin Shah, and Carroll Wainwright for their feedback and ideas.

There's a good chance I'd like to publish this at some point as part of a larger work. However, I wanted to make the work available now, in case that doesn't happen soon. 

They can't prove the conspiracy... But they could, if Steve runs his mouth. 

The police chief stares at you.

You stare at the table. You'd agreed (sworn!) to stay quiet. You'd even studied game theory together. But, you hadn't understood what an extra year of jail meant. 

The police chief stares at you.

Let Steve be the gullible idealist. You have a family waiting for you.

 

Sunlight stretches across the valley, dappling the grass and warming your bow. Your hand anxiously runs along the bowstring. A distant figure darts between trees, and your stomach rumbles. The day is near spent. 

The stags run strong and free in this land. Carla should meet you there. Shouldn't she? Who wants to live like a beggar, subsisting on scraps of lean rabbit meat? 

In your mind's eye, you reach the stags, alone. You find one, and your arrow pierces its barrow. The beast bucks and bursts away; the rest of the herd follows. You slump against the tree, exhausted, and never open your eyes again.

You can't risk it.

People talk about 'defection' in social dilemma games, from the prisoner's dilemma to stag hunt to chicken. In the tragedy of the commons, we talk about defection. The concept has become a regular part of LessWrong discourse. 

Informal definition. A player defects when they increase their personal payoff at the expense of the group.

This informal definition is no secret, being echoed from the ancient Formal Models of Dilemmas in Social Decision-Making to the recent Classifying games like the Prisoner's Dilemma

you can model the "defect" action as "take some value for yourself, but destroy value in the process". 

Given that the prisoner's dilemma is the bread and butter of game theory and of many parts of economics, evolutionary biology, and psychology, you might think that someone had already formalized this. However, to my knowledge, no one has.

 

Formalism

Consider a finite -player normal-form game, with player  having pure action set  and payoff function . Each player  chooses a strategy  (a distribution over ). Together, the strategies form a strategy profile  is the strategy profile, excluding player 's strategy. A payoff profile contains the payoffs for all players under a given strategy profile.

A utility weighting  is a set of  non-negative weights (as in Harsanyi's utilitarian theorem). You can consider the weights as quantifying each player's contribution; they might represent a percieved social agreement or be the explicit result of a bargaining process. 

When all  are equal, we'll call that an equal weighting. However, if there are "utility monsters", we can downweight them accordingly. 

We're implicitly assuming that payoffs are comparable across players. We want to investigate: given a utility weighting, which actions are defections? 

Definition. Player 's action  is a defection against strategy profile  and weighting   if 

  1. Social loss: 

If such an action exists for some player , strategy profile , and weighting, then we say that there is an opportunity for defection in the game.

Remark. For an equal weighting, condition (2) is equivalent to demanding that the action not be a Kaldor-Hicks improvement

Payoff profiles in the Prisoner's Dilemma. Red arrows represent defections against pure strategy profiles; player 1 defects vertically, while player 2 defects horizontally. For example, player 2 defects with  because they gain () but the weighted sum loses out ().

Our definition seems to make reasonable intuitive sense. In the tragedy of the commons, each player rationally increases their utility, while imposing negative externalities on the other players and decreasing total utility. A spy might leak classified information, benefiting themselves and Russia but defecting against America. 

Definition. Cooperation takes place when a strategy profile is maintained despite the opportunity for defection.

Theorem 1. In constant-sum games, there is no opportunity for defection against equal weightings.

Theorem 2. In common-payoff games (where all players share the same payoff function), there is no opportunity for defection.

Proposition 3. There is no opportunity for defection against Nash equilibria.

An action  is a Pareto improvement over strategy profile  if, for all players ,.

Proposition 4. Pareto improvements are never defections. 

Game Theorems

We can prove that formal defection exists in the trifecta of famous games. Feel free to skip proofs if you aren't interested.

In (a), variables stand for emptation, eward, unishment, and ucker. A  symmetric game is a Prisoner's Dilemma when . Unsurprisingly, formal defection is everywhere in this game.

Theorem 5. In  symmetric games, if the Prisoner's Dilemma inequality is satisfied, defection can exist against equal weightings.

Proof. Suppose the Prisoner's Dilemma inequality holds. Further suppose that . Then . Then since  but , both players defect from  with .

Suppose instead that . Then , so . But , so player 1 defects from  with action , and player 2 defects from  with action . QED.


 symmetric game is a Stag Hunt when . In Stag Hunts, due to uncertainty about whether the other player will hunt stag, players defect and fail to coordinate on the unique Pareto optimum . In (b), player 2 will defect (play ) when . In Stag Hunts, formal defection can always occur against mixed strategy profiles, which lines up with defection in this game being due to uncertainty.

Theorem 6. In  symmetric games, if the Stag Hunt inequality is satisfied, defection can exist against equal weightings.

Proof. Suppose that the Stag Hunt inequality is satisfied. Let  be the probability that player 1 plays . We now show that player 2 can always defect against strategy profile  for some value of .

For defection's first condition, we determine when :

This denominator is positive ( and ), as is the numerator. The fraction clearly falls in the open interval .

For defection's second condition, we determine when

Combining the two conditions, we have

Since , this holds for some nonempty subinterval of . QED.


 symmetric game is Chicken when . In (b), defection only occurs when : when player 1 is very likely to turn, player 2 is willing to trade a bit of total payoff for personal payoff.

Theorem 7. In  symmetric games, if the Chicken inequality is satisfied, defection can exist against equal weightings.

Proof. Assume that the Chicken inequality is satisfied. This proof proceeds similarly as in theorem 6. Let  be the probability that player 1's strategy places on .

For defection's first condition, we determine when :

The inequality flips in the first equation because of the division by , which is negative ( and ). , so ; this reflects the fact that  is a Nash equilibrium, against which defection is impossible (proposition 3).

For defection's second condition, we determine when

The inequality again flips because  is negative. When , we have , in which case defection does not exist against a pure strategy profile.

Combining the two conditions, we have

Because ,

QED.

Discussion

This bit of basic theory will hopefully allow for things like principled classification of policies: "has an agent learned a 'non-cooperative' policy in a multi-agent setting?". For example, the empirical game-theoretic analyses (EGTA) of Leibo et al.'s Multi-agent Reinforcement Learning in Sequential Social Dilemmas say that apple-harvesting agents are defecting when they zap each other with beams. Instead of using a qualitative metric, you could choose a desired non-zapping strategy profile, and then use EGTA to classify formal defections from that. This approach would still have a free parameter, but it seems better.

I had vague pre-theoretic intuitions about 'defection', and now I feel more capable of reasoning about what is and isn't a defection. In particular, I'd been confused by the difference between power-seeking and defection, and now I'm not.

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Planned summary for the Alignment Newsletter:

We often talk about cooperating and defecting in general-sum games. This post proposes that we say that a player P has defected against a coalition C (that includes P) currently playing a strategy S when P deviates from the strategy S in a way that increases his or her own personal utility, but decreases the (weighted) average utility of the coalition. It shows that this definition has several nice intuitive properties: it implies that defection cannot exist in common-payoff games, uniformly weighted constant-sum games, or arbitrary games with a Nash equilibrium strategy. A Pareto improvement can also never be defection. It then goes on to show the opportunity for defection can exist in the Prisoner’s dilemma, Stag hunt, and Chicken (whether it exists depends on the specific payoff matrices).

As others have mentioned, there's an interpersonal utility comparison problem. In general, it is hard to determine how to weight utility between people. If I want to trade with you but you're not home, I can leave some amount of potatoes for you and take some amount of your milk. At what ratio of potatoes to milk am I "cooperating" with you, and at what level am I a thieving defector? If there's a market down the street that allows us to trade things for money then it's easy to do these comparisons and do Coasian payments as necessary to coordinate on maximizing the size of the pie. If we're on a deserted island together it's harder. Trying to drive a hard bargain and ask for more milk for my potatoes is a qualitatively different thing when there's no agreed upon metric you can use to say that I'm trying to "take more than I give".


Here is an interesting and hilarious experiment about how people play an iterated asymmetric prisoner's dilemma. The reason it wasn't more pure cooperation is that due to the asymmetry there was a disagreement between the players about what was "fair". AA thought JW should let him hit "D" some fraction of the time to equalize the payouts, and JW thought that "C/C" was the right answer to coordinate towards. If you read their comments, it's clear that AA thinks he's cooperating in the larger game, and that his "D" aren't anti-social at all. He's just trying to get a "fair" price for his potatoes, and he's mistaken about what that is. JW, on the other hand, is explicitly trying use his Ds to coax A into cooperation. This conflict is better understood as a disagreement over where on the Pareto frontier ("at which price") to trade than it is about whether it's better to cooperate with each other or defect.

In real life problems, it's usually not so obvious what options are properly thought of as "C" or "D", and when trying to play "tit for tat with forgiveness" we have to be able to figure out what actually counts as a tit to tat. To do so, we need to look at the extent to which the person is trying to cooperate vs trying to get away with shirking their duty to cooperate. In this case, AA was trying to cooperate, and so if JW could have talked to him and explained why C/C was the right cooperative solution, he might have been able to save the lossy Ds. If AA had just said "I think I can get away with stealing more value by hitting D while he cooperates", no amount of explaining what the right concept of cooperation looks like will fix that, so defecting as punishment is needed.

In general, the way to determine whether someone is "trying to cooperate" vs "trying to defect" is to look at how they see the payoff matrix, and figure out whether they're putting in effort to stay on the Pareto frontier or to go below it. If their choice shows that they are being diligent to give you as much as possible without giving up more themselves, then they may be trying to drive a hard bargain, but at least you can tell that they're trying to bargain. If their chosen move is conspicuously below (their perception of) the Pareto frontier, then you can know that they're either not-even-trying, or they're trying to make it clear that they're willing to harm themselves in order to harm you too.

In games like real life versions of "stag hunt", you don't want to punish people for not going stag hunting when it's obvious that no one else is going either and they're the one expending effort to rally people to coordinate in the first place. But when someone would have been capable of nearly assuring cooperation if they did their part and took an acceptable risk when it looked like it was going to work, then it makes sense to describe them as "defecting" when they're the one that doesn't show up to hunt the stag because they're off chasing rabbits.

"Deliberately sub-Pareto move" I think is a pretty good description of the kind of "defection" that means you're being tatted, and "negligently sub-Pareto" is a good description of the kind of tit to tat.

As others have mentioned, there's an interpersonal utility comparison problem. In general, it is hard to determine how to weight utility between people.

I actually don't think this is a problem for the use case I have in mind. I'm not trying to solve the comparison problem. This work formalizes: "given a utility weighting, what is defection?". I don't make any claim as to what is "fair" / where that weighting should come from. I suppose in the EGTA example, you'd want to make sure eg reward functions are identical.

"Deliberately sub-Pareto move" I think is a pretty good description of the kind of "defection" that means you're being tatted, and "negligently sub-Pareto" is a good description of the kind of tit to tat.

Defection doesn't always have to do with the Pareto frontier - look at PD, for example.  are usually all Pareto optimal. 

It's worth being careful to acknowledge that this set of assumptions is far more limited than the game-theoretical underpinnings. Because it requires interpersonal utility summation, you can't normalize in the same ways, and you need to do a LOT more work to show that any given situation fits this model. Most situations and policies don't even fit the more general individual-utility model, and I suspect even fewer will fit this extension.

That said, I like having it formalized, and I look forward to the extension to multi-coalition situations. A spy can benefit Russia and the world more than they hurt the average US resident.

I very much agree that interpersonal utility comparability is a strong assumption. I'll add a note.

Combining the two conditions, we have

Since , this holds for some nonempty subinterval of .


I want to check that I'm following this. Would it be fair to paraphrase the two parts of this inequality as:

1) If your credence that the other player is going to play Stag is high enough, you won't even be tempted to play Hare.

2) If your credence that the other player is going to play Hare is high enough, then it's not defection to play Hare yourself.

?

I guess the rightmost term could be zero or negative, right? (If the difference between T and P is greater than or equal to the difference between P and S.) In that case, the payoffs would be such that there's no credence you could have that the other player will play Hare that would justify playing Hare yourself (or justify it as non-defection, that is).

So my claim #1 is always true, but claim #2 depends on the payoff values.

In other words, Stag Hunt could be subdivided into two games: one where the payoffs never justify playing Hare (as non-defection), and one where they sometimes do, depending on your credence that the other player will play Stag.

Yes, this is correct. For example, the following is an example of the second game: