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).

 

----------------------------------------------

 

 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.

 

----------------------------------------------

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