She laughed again, encouraged by Hume’s obvious fascination.
‘In other words, it’s the very opposite of the lack of communication that the prisoners had in their cells or you had to contend with when you were made to sit behind the book barrier.’
‘So, what do you think the implications of this thinking are for the Dilemma, Sophie?’ asked Hume, now intrigued to see if her conclusions fitted with his earlier insight. ‘Did you actually arrive at a strategy that would do this?’
Sophie waved her hand over the reams of workings.
‘Well, I’ve tried many different approaches but it’s only when you’ve played enough hands over a long period of time that the strengths and weaknesses of different strategies begin to show up. I’ve been playing against myself for hours, trying everything from ironclad discipline to a mixture of the prescribed and the random. But, yes, I do believe that I’m making progress. I wondered early on if the simple choice of always co-operating would not bring the most points – in the way that the saintly hope they will bring round the wrongdoer. But I came to the view that once selfish people see that they can defect with these committed co-operators without any loss to their own points then the temptation to take advantage of their good natures would be too great for opportunists to resist. Then, for a time I had great hopes for the tactic of simply repeating the same choice if one won but changing it if one lost. It produced promising results for a time but then began to disappoint. However, I persisted with the underlying aim of following one’s opponent’s choice, and yes, there is indeed one strategy that keeps winning.’
‘How very interesting Sophie,’ said Hume, now completely captivated, ‘what is it?’
‘ Well, I think of it as Tit for Tat,’ she replied, ‘and it works by simply repeating the choice of the other person. So, when the other person defects I play ‘defect’ next. When he co-operates I do the same. It seems to me that it wouldn’t take long for him to realise what’s happening. In fact the whole point of the Tit for Tat strategy is that it shouldn’t be a secret. You actually want the other person to know what you’re doing.
‘And it turns out to be a very powerful way of playing, just because it’s so transparent – it’s so easy for your opponent to understand. Its strength comes from an irresistible combination of trust, retaliation, forgiveness and clarity. By ‘trust’ I mean that the other person can trust you to play co-operate if he does. If he doesn’t – and instead plays defect – then he knows that you will immediately retaliate. And this discourages him from persisting whenever defection is tried. The reason is obvious – why would he want to get just one point when he can see that three are available?
‘Once he stops playing defect and goes back to co-operate you restore co-operate as well, so it’s a kind of immediate forgiveness. And the strategy’s clarity makes it intelligible to the other player, because what you’re doing is so evident – and it therefore achieves long term co-operation.’
‘But Sophie,’ said Hume, excitedly, ‘did no other strategy achieve the same end?’
‘No.’ Sophie shook her head. ‘At first, of course, the nasty, defector strategies kept winning at the expense of the naïve, co-operative players. Only retaliator strategies like Tit for Tat where you have a co-operator who will immediately switch to defect if the other person defects, were able to resist them. But gradually the nasty approaches ran out of the easy pickings of killing co-operators and then defectors kept meeting each other – and obviously their numbers began to decline. This is really the key point. If the defectors kill the co-operators then they’re left having to deal with each other! They then dwindle because they kill each other. This was when one sees the success of the Tit for Tat strategy because its retaliation mechanism eventually takes command of the game. Do you see now why I say that there must be a way of playing that educates defectors and rewards co-operators?
‘The way it works shows itself over time – it may lose or draw each battle, but it wins the war. It ensures that most of its games are high scoring encounters and so it constantly brings home the most points. The crucial insight about the Tit for Tat strategy is that it isn’t trying to ‘beat’ its opponent – success need not be at the price of someone’s failure. A nil sum game is the last thing you want. Tit for Tat is an agreement not a contest.’
* * *
It was now mid morning and Zweig was in agony, his muscles screaming at their continued contraction. For a bad few minutes he hadn’t even known whether he could continue. Then he summoned up a vision of Sophie and yet again he demanded a further effort from his tortured body. He knew there was no going back now, that this was a fight to the end and that his whole being would never allow himself to fail. He leant forward once more, forcing himself to focus his pain on the image of Dunbeath’s sneering face that day on the dunes. Slowly his mind conquered the shrieking muscles and again he settled down for the battle with himself that he knew would have to be won.
* * *
‘Of course,’ Sophie continued as Hume listened, increasingly intrigued by her unfolding logic, ‘the principal requirement for Tit for Tat to work is for two people to have a stable, repetitive relationship. In fact, the longer a pair of individuals interact, the greater the chances of co-operation. On the other hand, the more casual and opportunistic an encounter, the less likely it is that Tit for Tat will succeed in building co-operation. New players are wary. They don’t know if they are in a one-time contest like the desperate choice in Lord Dunbeath’s original Prisoner’s Dilemma or whether they are embarking on a longer-term relationship. And, if they are, are they dealing with a serial defector or with someone who would wish to share in a co-operative relationship?’
David Hume sat quite still for a few seconds, turning this over in his mind.
‘I can only repeat that I find this most revealing, Sophie,’ he said eventually, before giving a slight grimace as his thoughts alighted on an objection. ‘But, of course, there is much more subtlety in this problem than the Dilemma allows. There must be a great difference, for example, between a ‘bad’ defector who ruthlessly defects as a matter of policy and personality and a ‘good’ defector who’s defecting only to protect himself. I’m afraid that Lord Dunbeath, for example, has shown himself to be unable to break free of the Urquhain obsession of never trusting anyone. But there are many other people who would defect out of a natural sense of self-preservation and who would actually prefer to co-operate. Still, I think your Tit for Tat strategy is an extraordinary insight. It’s exactly the kind of empirical solution that could show us how our behaviour towards each other has evolved.
‘If the Prisoner’s Dilemma is, as you say, just a building block in showing how a society can function then I’d suggest we have to imagine how it works in situations closer to reality – with lots of players rather than just two. And each of them playing with all of the others. The possibilities of all these multiple Dilemmas make the head spin. I’ve no doubt that as my friend, Adam Smith, would say about markets, some invisible grinding force would come into play eventually to restore equilibrium.’
He rose to his feet and walked around the room, deep in thought. Sophie followed him with her eyes, quite certain that he was about to bring a further twist to what she had outlined.
‘But if you’re right about Tit for Tat – and I think you are – then everything makes sense,’ he said at last.
‘In what way, Mr Hume?’
‘Well,’ Hume replied, eager now to take Sophie’s conclusions further, ‘if you think of defectors as hawks and co-operators as doves then the hawks would easily keep defeating the doves – as Dunbeath did to me – until they were killed, or went back to Edinburgh, or there were so few of them left that the hawks kept meeting others like themselves. They then kill or negate the other hawks because they know no other way. This is very much as you’ve just said.
‘However, say there are what you called retaliators - doves that fight back and behave like hawks – then th
e hawks would decline to them because the retaliators would work in teams to succeed. They know how to share. We see this in societies that become threatened, don’t we? Where good, peace loving people reach a point where they say they are ‘standing up for what they believe in’ and suddenly become surprisingly warlike. But, going back to my parable, when the hawks have been defeated or change their ways and the retaliators turn back into co-operative doves then the process would end with the evolution of a stable, co-operative, three point gaining society. I’m not saying that there need be a rapid progress in this but I do believe that it is one that would have an inevitable outcome.
‘In other words although the possibility of retaliation remains as a strategy, after a bit the players in a co-operative relationship begin to look as if they are no longer playing Tit for Tat but instead have chosen what we might term Always Co-operate – and are always receiving three points. But a defector would find that this is untrue if he tried to free ride, to grab an advantage at their expense.’
Sophie was about to reply to this when the door burst open and Dunbeath flew angrily into the room. He glared at them as if they were somehow at fault and then went over to the window, shouting loudly as he did so.
‘That stupid ship’s captain is still sitting out there, Hume! He is beginning to irk me with his constant staring. What is he looking at? What does he want? Much more of this and I shall shoot him! I wouldn’t be surprised if I found he was within range from this very window.’
Sophie immediately tried to deflect their attention from Zweig and to defuse the situation.
‘Will you not join us in our discussion, my lord? Mr Hume and I are making great progress in finding a mathematical approach to the Dilemma. Would you like to hear about an interesting strategy we’ve arrived at?
Dunbeath managed to drag himself away from the window but just glowered at her.
‘What’s that you say, Sophie?’ he said testily, clearly struggling to control his anger. ‘The Dilemma? No, no I have no time for that. There is still much to do to prepare for the Board of Longitude. It must be soon. It must be. Damn it! Why haven’t I been informed of a date yet?’
* * *
Major Sharrocks walked slowly through Craigleven’s great staterooms towards L’Arquen’s office. His progress was little more than a crawl and he shot an exasperated glance at the soldier alongside him. The trooper was aware of the major’s irritation and he gave another shove to the bent figure that shuffled so slowly ahead of him. The man could barely walk, bound at the ankles and with his hands tied behind his back. Sharrocks looked with disgust at the torn tweed of his kilt and his mud caked face. By God, these people were repulsive, he thought and wondered yet again why on earth he was here, trying to control such barbarians. For the life of him he couldn’t see why anyone would want to have these filthy Scots as part of the kingdom.
Eventually they reached L’Arquen’s beautiful walnut door. At first there was no answer to Sharrocks’ knock and he banged on the door for a second time. He turned to the guard.
‘You’re quite sure he’s in there?’
‘Yes, sir. Without a doubt sir,’ said the man with a flush of embarrassment.
Sharrocks turned to the trooper he had with him.
‘You wait here with this revolting specimen,’ he said, nodding towards the highlander. ‘I’ll call for you in a minute.’
Sharrocks knocked a third time, more loudly than before, and pushed the door gradually open. Peering around it he saw L’Arquen lounging with his feet on the desk, his eyes closed, a mass of papers carelessly thrown in front of him.
‘Our noon meeting sir?’ said Sharrocks, refusing to be drawn into L’Arquen’s absurd power game.
The colonel opened his eyes at last.
‘Mmm? Ah Sharrocks, there you are. Is it midday already? So soon? Well, come in, come in, what have you to report?’
‘You will recall that there was a man on the dunes in Dunbeaton Bay yesterday, sir, staring at the Castle of Beath. He is still there. We observed him all night and he did not give up his position for so much as a second.’
‘How very curious, Sharrocks,’ replied L’Arquen, now moving to sit upright in his chair, ‘but it may be nothing. Keep him under surveillance – but covertly. I have told you many times, I do not want it known that we’re keeping watch. As I say, it may be nothing.
‘Speaking of Dunbeaton, I have something for you, major. Our people have reported back that the ship that blew up, the Schwarzsturmvogel, had sailed from Königsberg. It is in Prussia, Sharrocks. Apparently it was bound for the quarrying trade in France. I have the captain’s name somewhere.’ L’Arquen leant forward and rummaged about his papers. He drew out a military signal. ‘Yes, here it is. His name was Alexis Zweig. No doubt he was among the corpses you saw. Still, I have sent word back that they are to make further investigations. So, what else do you have?’
‘We have a suspect sir. Found near Lanochburn. You’ll remember that we had reports of rebels in the area and we’ve been keeping an eye on the place for some days. He was hiding in a hay barn. Refuses to say anything, even to give his name. We have him outside your door.’
‘Very good, Sharrocks. Bring in him. We shall have a quiet word with him.’
The major opened the door and beckoned to the trooper. He came into the room, pushing the prisoner before him and then set the man in front of the colonel’s desk. L’Arquen stood up, his face wreathed in smiles.
‘My dear sir. How very good of you to give us your time. I must apologise if you’ve been brought here against your will but you’ll know that we live in uncertain times. Now, I’d be very grateful if you would tell me where your friends are. We must find these rebels of yours and stop their nonsense before they can do any more harm. Would you be kind enough, please, to tell me where they have hidden themselves?’
The man lifted his eyes from the floor and glared at L’Arquen, hatred scored in every line of his face. He said nothing. After a few seconds the colonel broke the silence.
‘No answer? I fear that does not speak well of you, sir. If you have no part in all this then I would simply like you to tell me that.’
There was a further tense, black pause.
‘Harken,’ said L’Arquen softly, his voice barely more than a whisper. The room froze, the soldiers only too aware that a terrible abyss had been reached. Not for nothing was the colonel known to his men as ‘Harken L’Arquen’.
‘Harken,’ he said for a second time, again very quietly, ‘I would greatly appreciate it if you would tell me what you know.’
The highlander stood quite still, his mouth a tight line. L’Arquen gave a low sigh.
‘Very well. I quite understand. You have nothing to say. Well now, my dear sir, this gentleman here is called Trooper Williams and he will take you away again,’ the colonel waved his arm towards the soldier, ‘and if you should change your mind and want to come back to see me later then you will always be most welcome.’
The trooper manhandled the prisoner out of the office and L’Arquen rose and went over to the map. He looked at it closely, trying to find Lanochburn, and then stood for a time with his finger on it as if considering the implications of the rebels’ position. In the distance, possibly in a room behind the office, someone began to scream. At first there was a note of surprise in its tone but that quickly gave way to a kind of indignation, then anger, pleading and finally to an animal shriek of such pure pain, so wild and so all consuming that there was no human thinking left in it.
Sharrocks winced at the sound. He might loathe the Scots but this was not his way. He looked at L’Arquen with misgiving.
But the colonel continued as if he hadn’t heard anything.
‘Your little play on the dunes intrigues me, Sharrocks. There is plainly some heat in all this and I rather suspect the drama may come to the boil soon. Report back to me if the play has not moved on by tomorrow.’
* * *
That night David Hume sat
at the desk in his bedroom writing a letter by candlelight. A slight smile played over his lips as he wrote. He knew he had to be brief as Annie would be up at dawn to take the trap to Wick and he wanted the letter to be with her to deliver to the mail coach for Edinburgh. He smiled again as he thought of his new friend Adam Smith and what he would make of their discussions at the castle. Briefly he described the progress they were making with the game:
…I have written to tell you before of Miss Kant. She is the most remarkable woman I believe I have ever met. What a mind! She is a mathematician and she has set herself to play game after game of the Prisoner’s Dilemma looking for the best way of accumulating points by encouraging long term co-operation. She has arrived at a solution that is so elegant that I have no doubt it has the inevitability of truth about it.
I have so much to tell you when we next meet but, briefly, the results are leading me to a conclusion on the reasons for trust and the nature of benevolence that’s far beyond anything I described in the Treatise.
It is nothing less than the realisation that the reason we are ever virtuous with each other is because we instinctively know that it is a winning strategy. If you want to win in life you have to be selfish yet disguise your hand in the pretence that you are not. And add to this the key ingredient of irrationality – trust.
In other words, Mr Smith, the mathematical conclusion the Dilemma leads us to is something that I have written about in part but never so completely. It is that the origin of all virtue is selfishness. Yes – all!
It is because we are calculating, selfish beings whose base instinct is to survive and prosper that we are ever virtuous. Is this not then the linkage we sought between my belief in man’s natural benevolence, the altruism and compassion we see in people that we admire so much and of which we spoke – and your mechanism of self-interested specialisation? Is this not the solution to the problem I put to you when we first met? That ties our base drives to our inexplicably sympathetic natures?
The Prisoner's Dilemma Page 18
