Ethical Farmers and Zero Sum Games

In A Treatise of Human Nature, David Hume described a scenario involving two neighbouring farmers which closely matched an asynchronous prisoners’ dilemma – but apparently without the equivalent of a prosecutor.  If you haven't already, please see Ethical Prisoners for the ramifications of having a prosecutor involved - the conclusion to this article will probably not make much sense if you haven't read the preceding article.

Each of Hume's farmers may choose to spend one day helping the other to reap his harvest and minimise wastage in the fields, or commit to working independently and accept the loss represented by such wastage.  The problem is that they have to work on one farm one day and on the other farm the other day.

For this to work, the first farmer (let's call him Cain) must trust his neighbour (Abel) to make good on a promise to help while he himself actually commits to work on Abel’s field.  It seems that there is no-one for Abel to compete with except Cain, so rationally, Abel should betray Cain’s trust.  However, Cain should expect that and should therefore never extend that trust in the first instance.  Rationally it seems, Cain and Abel should work their own fields and accept inevitable losses, rather than expend effort on helping their neighbour.

Real-world examples indicate that this does not occur, but does this mean that farmers are irrational with regard to the trust that they extend to each other?  A reconsideration of the scenario shows that this is not the case.  While there is no human analogue of the prosecutor, farmers are very usually acutely aware that their major adversary is not their neighbour but rather nature herself.  Farmers don’t play against each other – they play against nature, fighting against weather, time and the very earth from which they wring their living.

When this is taken into account, it becomes entirely rational that farmers co-operate because if they don’t, nature will beat them both every time.

It should be noted that the situation with Hume’s farmers is not only asynchronous, but they seek to maximise a gain rather than minimise a penalty.  This can of course be reworded such that each collecting a full harvest is equivalent to a Minimum Punishment while both suffering waste because they both fail to co-operate is equivalent to Medium Punishment for both. 

If one farmer decides to co-operate with the other, and the other decides not to co-operate, then the co-operative farmer will lose time that could otherwise be devoted to his own harvesting.  A co-operative farmer with an unco-operative neighbour ends up in a worse situation than if he had been unco-operative as well – an equivalent to Maximum Punishment.  On the other hand, the unco-operative neighbour obtains a full harvest and doesn’t have to expend any extra effort – which is equivalent to being set free.

 

Considered thus it can be seen that co-operation is equivalent to staying silent and refusal to co-operate (defection) is the same as confessing, as can be seen by comparing the following with Larry’s first consideration in the previous article (Ethical Prisoners):

Neither the prisoners’ dilemma framed in the previous article, nor Hume’s farmer scenario, are truly zero-sum games.  In a zero sum game the "players" play within a closed system as it were – so if one player wins it is always at the expense of another player.  Imagine instead a game in which each player competes to turn the tiles on a board to their own colour.

Initially let us say that 32 tiles are blue and 32 tiles are yellow.  Each player can only prevail by taking tiles from the other player.  It is not possible to co-operate and thus create more tiles and failing to co-operate does not reduce the number of tiles available.  The game is "zero-sum" because the end result will be (32+X) tiles to the winner and (32–X) tiles to the loser, (X) + (–X) = 0.  So, the sum of the gains and losses within the game is zero.

Beyond actual games, it is exceptionally difficult to conceive a real life situation in which a ‘game’ is truly zero sum.  This is most probably because a zero sum game is played within a closed system and there are no closed systems in real life.  Even so, as it is possible that either heavily disguised zero sum games or games which are effectively zero sum exist in reality, we shall briefly look at the considerations that a player within such a game must make.

As a player can only win at the expense of another player, mutual co-operation or mutual refusal to co-operate (defection) result in no change in relative position.  Therefore, considering a game of coloured tiles:

 



 

The rational decision for the yellow player is to choose not to co-operate.  In a zero sum game there is no third player, overt or otherwise; it is inherent in the game that either one player wins and the other player loses or they draw.  Even if there were to be a third player outside the game who would lose if the number of blue and yellow tiles remained equal, that outcome can be achieved if both players consistently refuse to co-operate, without the need for either of the players inside the game to risk losing against the other by exposing themselves to the risk of betrayal.

It would be entirely irrational for any player in a zero sum game to trust the other player or to risk future betrayal by responding favourably to any trust extended to them.

It may be beneficial to compare the zero sum game with the prisoners’ dilemma.  If the prosecutor were not as clever as she might otherwise be, she might choose to offer the prisoners an alternative deal in which the Minimum Punishment and the Medium Punishment were the same, perhaps both 10 years.  Initially this seems to be in her favour because the range of outcomes now consists of: one conviction resulting in a Serious Punishment of 20 years, or two convictions resulting in two awards of Medium Punishment of 10 years.  She stands to secure 20 years of jail time either way.

Apparently she can’t lose.

This conclusion is, however, based on the assumption that the prosecutor doesn’t care how she distributes the 20 years of jail time.  This seems unreasonable in terms of the scenario since she could just put both in front of a court and not care whether they confess or not.  Her best outcome is to not make the offer in the first place and hope that at least one confesses on the stand – condemning both prisoners and resulting in two 20 year sentences.  By virtue of the fact that the prosecutor did in fact make the offer, we must conclude that for some reason she does care how she distributes the 20 years of jail time that she can be assured of.  She cares so much that she is willing to forego the possibility of securing two 20 year sentences in favour of what she calculates as a greater chance of securing one 20 year sentence.  It’s so important to her that, in the standard framing of the dilemma, she is even willing to risk only securing two 5 year sentences.

What could we expect if the prosecutor were to offer only two options: shared Medium Punishment for confess-confess or silent-silent outcomes; or Severe Punishment for one and freedom for the other for confess-silent outcomes? 

 

With this offer, the prosecutor would significantly reduce the likelihood of one of the prisoners staying silent.  Under these circumstances, a prisoner could only ever lose if he were to stay silent while the other confesses.  He cannot win against the prosecutor because the option of shared Minimum Punishment is not offered – the prosecutor wins (secures 20 years of jail time) no matter what choice the prisoners make.  Therefore the prisoners find themselves in a zero-sum game such that they can only play against each other so, given that they are rational actors and only stand to lose if they stay silent, they must rationally both confess.

If as rational actors they must both confess, then the prosecutor may be tempted to increase the Medium Punishment for bilateral confession, for example to 15 years each for a total of 30 years.  This, however, changes the dynamics once again – oddly enough giving control to the prisoners.  Now each prisoner has control over the game and can avoid a loss, depending on his chosen opponent.  Larry can choose to win against the prosecutor by staying silent which would deny her her best outcome of 30 years served (either through serving 10 years in prison together with Wally or 20 years alone, if Wally were to choose to play against Larry).  Alternatively, Larry can choose to ensure that he does not lose against Wally, which he can acheive by confessing, resulting in either 15 years in prison together with Wally (a draw) or freedom (a win).

 

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 Again it seems that the primary consideration in these games is "who are you playing against?"  We'll be returning to these issues in a later article.

 

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This article is one of a series.  It was preceded by Ethical Prisoners and is followed by Morally Circular Definitions.

Ethical Prisoners

The prisoners’ dilemma is often described, approximately, as follows:

Two people have been arrested for a serious crime and since been separated such that they cannot communicate.  The prosecutor provides an ultimatum to both of them individually:

You may choose to either confess or remain silent.

If you both choose to remain silent, rest assured that we have enough evidence to convict you both of a crime, albeit one which is less serious and attracts a lesser sentence than that for which you have been arrested.

If you both confess, you will both be charged with the crime for which you have been arrested, I will reward you both with a reduction in your sentences in recognition of that.

If you confess while the other remains silent, I will set you free.  I will use your evidence to ensure that the other will serve the maximum sentence possible.

This scenario clearly has ethical implications, but which ethical principle has precedence is not so clear.  There are also a number of uncertainties which I have glossed over which makes determining precedence extremely difficult.  Are the two people long-term friends or strangers?  Did they actually commit the crime?  Is the prosecutor reliable, that is will the prosecutor keep her word? 

What is the extent of the difference between the punishments?  Is the death sentence involved?  Are they part of an organisation which will punish a confessor on release?  What are the ethical stances of the prisoners – is lying worse than abandoning a colleague or vice versa? 

It is possible that neither prisoner will be aware whether the other has been given this ultimatum yet, or indeed ever will be.  From that perspective, it can be argued that the single prisoner may only be playing a game with the prosecutor, not the other prisoner.  Otherwise, in reality, the prisoner is playing two different games simultaneously – one with the other prisoner and one with the prosecutor.

In the standard treatment of the prisoners’ dilemma the following is assumed:

·         Both prisoners act rationally, based on nothing more than the information provided. 

·         Both prisoners consider only the best possible (short term) outcome for themselves without fear of ethical constraints or retribution after release.

·         Both prisoners may consider the other as little more than an abstraction, so no consideration as to the welfare of the other is necessary.  Both are aware that the other was also taken into custody.

·         The crime in question is sufficiently mundane that no lasting stigma is attached to admitting to it (at least from the perspective of the prisoners).

·         The rules of the game are fair and thus the prosecutor is not lying, provides both prisoners with the same dilemma and will keep her word.

Furthermore, the scale of punishments is such that the punishment accorded to a betrayed prisoner, one who remains silent while the other confesses, is worse than any punishments meted out to both.  We’ll call this the “Severe Punishment” and arbitrarily set it to 20 years in prison.   The punishment which would be shared by both prisoners is significantly lighter if both stay silent than if both confess.  We’ll call the former the “Minimum Punishment” (5 years in prison) and the latter the “Medium Punishment” (10 years in prison).  The reward for unilateral confession, freedom and immunity from prosecution or “No Punishment”, is preferable to all other outcomes.  (Variations of these scales of punishment are also studied by games theorists, but we shall not delve quite so deeply.)

The problem facing each prisoner is that the reward or punishment to be accorded is not entirely of his own making.  Consider the decision making process only one prisoner, calling him Larry for convenience.  Larry has two choices, to stay silent or to confess.  Larry knows that his partner in crime (Wally) has the same two choices, but he doesn’t know for sure what Wally will do and he knows that Wally doesn’t know what he, Larry will do.

If Larry chooses to stay silent, two outcomes are possible: Wally stays silent and they both receive the Minimum Punishment of 5 years or Wally confesses and is released while Larry receives the Severe Punishment of 20 years.

If Larry chooses to confess, two outcomes are possible: Wally stays silent and receives the Severe Punishment of 20 years while Larry is released or Wally confesses and both receive the Medium Punishment of 10 years.

What should Larry do?

Because Wally and Larry’s decisions are effectively simultaneous, Larry can consider his predicament from two perspectives: if Larry makes a certain decision, what will be the potential consequences of Wally’s decision? and, if Wally makes a certain decision, what options are open to Larry?

The first consideration can be illustrated thus:

Either

	Larry decides to confess
Wally decides to confess	10 years in jail for both
Wally decides to remain silent	Larry goes free
	Wally spends 20 years in jail

or

	Larry decides to remain silent
Wally decides to confess	Larry spends 20 years in jail
	Wally goes free
Wally decides to remain silent	5 years in jail for both

 

The second consideration can be illustrated thus:

Either

	Wally confesses
Larry decides to confess	10 years in jail for both
Larry decides to remain silent	Wally goes free
	Larry spends 20 years in jail

or

	Wally remains silent
Larry decides to confess	Wally spends 20 years in jail
	Larry goes free
Larry decides to remain silent	5 years in jail for both

The distinction between to the two modes of decision making is that the first is predictive (in which Larry makes a decision and hopes that Wally makes a particular decision) whereas the latter is reactive (one in which Wally makes or is assumed to have made a decision and Larry must ensure that his decision obtains the best result).

Looking at the predictive consideration:

·         If Larry decides to confess, he stands to gain freedom at the expense of Wally or to spend 10 years in jail, along with Wally.  Clearly in this instance, Larry will hope that Wally remains silent.

·         If Larry decides to stay silent, he might only spend only 5 years in jail together with Wally.  However, if Wally was going to stay silent anyway, then Larry has made a suboptimal decision which would have given him freedom.  Additionally, by remaining silent, Larry has exposed himself to the risk of spending 20 years in jail.

In terms of prediction, it appears that Larry’s best option is to confess.

Looking at the reactive consideration:

·         If Wally confesses, then Larry can either confess and only spend 10 years in jail along with Wally, or stay silent and allow Wally to walk free while he himself spends 20 years in jail. 

·         If Wally stays silent on the other hand, then Larry can either confess and be released or stay silent and spend 5 years in jail along with Wally.

In terms of reaction, the benefits accruing from confession and irrationality of staying silent are even starker.

The “dilemma” revolves around the fact that if the two are going to spend time in jail together (which seems the likely outcome given that both are rational agents and both will rationally decide to confess) then it is rational that they collectively strive to spend as little time in jail as possible – which they could accomplish by remaining silent.  Therefore, it’s rational to confess and it’s rational to remain silent.

It is tempting to consider this scenario in terms of ethics, both from the perspective of allowing ethics to sway the decision making of the prisoners and also by analysing the scenario to see if a moral structure could be derived from the hypothetical dilemma.

If ethics were allowed to sway the decision making of the prisoners, it would be necessary to decide which standard ethical principle has precedence – “do not lie” or “do not abandon/betray your colleagues”?  To determine this we need to know more about the prisoners.  For instance, we don’t know whether the prisoners actually committed the crime of which they are accused.  While we stated that the game is ‘fair’ it was not made clear that the prisoners were guilty or not.

If the prisoners are career criminals, then it is probable that they will value unity over honesty.  If the prisoners are innocent (and more, do not know the other prisoner at all), it is probable that unity will not figure highly.  This does not bode well for a concept of universal morality, at least as derived from the prisoners’ dilemma, because the morality to be applied is clearly dependent on the situation.

What we do see, however, is that the morality of unity does have a benefit in this situation.  It is generally considered moral to extend and honour trust.  In this instance, bilateral morality will benefit both prisoners.  Many natural situations are analogous to the prisoners’ dilemma and the morality of unity does seem to come into play in these situations. 

An example is when two ancient warriors first meet, and choose neither to shield themselves nor to raise their weapons against the other.  By extending trust and honouring that (apparent) trust, each the warriors risk death if the other plans betrayal.  What they avoid is the need to fight immediately and risk being wounded in an unnecessary battle while simultaneously standing to gain by potentially making an ally.

In the prisoners’ dilemma we can see that the morality of unity can reap the benefits of co-operation in a situation in which choosing not to co-operate would result in a worse outcome for both.  Is this sufficient basis for a universal morality?

Let us briefly consider a couple of issues.

First, who are the prisoners playing against?  Is Larry playing against Wally, or against the prosecutor (potentially together with Wally)?  When the scenario is framed, it is stated that the prosecutor is ‘fair’ but it is plain that she may benefit from the situation, or potentially fail to benefit, depending on the outcome.  If she obtains two confessions, that could be considered a win, possibly her best outcome.  However, she may consider one major conviction with a full sentence served to be preferable – we can reason that this is quite likely otherwise she wouldn’t make such a generous offer.  Her worst outcome is to have both prisoners remain silent, thus obtaining only two nominal convictions.

If Larry and Wally confess and remain silent with equal probability (randomly, rather than rationally), the prosecutor has a 75% chance of getting a favourable outcome – either two confessions or one major conviction.  If Larry and Wally are independent rational actors playing against each other, then both will almost certainly choose to confess, a favourable outcome for the prosecutor.  It is only when they choose to act ethically (applying the morality of unity) that the prosecutor will lose.

Therefore, exactly who it is that Larry plays against should be a vital part of his considerations.  Larry can only win against the prosecutor by staying silent and hoping that Wally does so too.  If Larry chooses to play against Wally though, then he should rationally choose to confess.  If Wally confesses then a draw results between the prisoners and the prosecutor secures a minor win.  If Wally chooses to play against the “wrong” person (that is against the prosecutor, and not Larry), then he will stay silent and Larry will win by walking free while Wally spends the next twenty years in jail – and the prosecutor secures a major win.

Second, we should consider other variants of the prisoners’ dilemma.  As the dilemma has been framed, it is a single event and effectively synchronous.  Most real-world situations are neither isolated events nor entirely synchronous.  There are variations of the prisoners’ dilemma within the discipline of game theory in which this is taken into account.

If the prisoners’ dilemma is asynchronous, the decision making processes of each prisoner will match with Larry’s hypothetical considerations above.  The prisoner who makes the first move uses the predictive consideration while the other uses the reactive.  Both prisoners will therefore rationally choose to confess, unless the first works on the assumption that they are playing against the prosecutor and not each other.  The second prisoner then has the opportunity to betray the other or to co-operate in order to beat the prosecutor by staying silent.

If the specific form of the prisoners’ dilemma is a series of dilemmas, then the prisoners can look at the problem in a number of different ways.  If both prisoners have a tacit agreement that they are playing against the prosecutor, then they will consistently choose to stay silent.  However, if at least one prisoner decides that he is playing against the other prisoner then other strategies arise.

Let us say that one prisoner is a superiority seeker, who wants no more than to prevail over a player who plays the game under the same conditions as himself.  To prevail in a recurring prisoners’ dilemma, a superiority seeker must win at least one more round than the other prisoner.  This can be achieved either by confessing in the first and all subsequent rounds and hoping that his opponent remains silent at least once or by staying silent in the first round, thereby potentially setting up a trust relationship which later be abused for profit.

The first option seems to be clearly more rational as the superiority seeker stands to do no more than come equal if the other prisoner follows the same strategy and will win otherwise.  The second option is risky with no clear benefit because by acting to set up a situation of trust, all the superiority seeker does is make it possible for his opponent to be the betrayer, rather than the betrayed.  The strategy could also fail in the very first round, if the opponent chooses to repeatedly confess.

It therefore seems to make little difference whether a prisoners’ dilemma is a singular event or recurring, or synchronous or asynchronous.  Who a player considers to be the opponent seems to remain the single most important determining factor.

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This article is one of a series.  It was preceded by The Moral Animal and will be followed by Ethical Farmers.