Hume persisted with his question.
‘When you wrote to me in Edinburgh you said that you had recently witnessed an event that had made you invent the Prisoner’s Dilemma. I wondered what it was.’
Dunbeath’s stickiness immediately started to return.
‘What? Did I say as much? Well, there was nothing. I can’t imagine why I would have said that.’ There was a tense silence and Hume exchanged a quick glance with Sophie. Then she rose quietly from the table and started to pick up the used dishes to take back to the kitchen.
* * *
Early the following morning Zweig returned to the shoreline to finish shaping the branch with his knife. He’d shaved it to be quite straight, stepping down the width of the cylinder towards the end and then bringing out the final section in a bulge. Now he cut slits in the upper part and began to force small stones into them, making certain that they would be held in place as the sodden wood shrank. Eventually he held the finished piece up to the light and then bounced it in the palm of his hand to check its weight.
It will pass, he thought to himself – as long as it isn’t looked at too closely.
He took the oilskin cloth from his pocket that had covered the Domenico Salva. He carefully wrapped it around the wooden cylinder and then ran his hand over the cloth. Bundled up in this way it looked and felt like the lost telescope. He quickly tied it with the original string and then set off to walk back to Dunbeaton.
Chapter 13
It was two weeks later and Dunbeath and Sophie were alongside one another at the great table in the tower study. Dunbeath glanced up from their papers and picked up a book of ecliptic tables, now thankfully found under the huge mass of written calculations that Sophie had carefully classified, organised and arranged. He could scarcely believe the changes she’d wrought. For the first time in years he was no longer drowning in paper and he still found it strange to be able to find things instead of having to wade through the quagmire he’d allowed to build up. His hand moved to stroke his chin, pleased too that he’d finally found the time to be clean shaven and the will to attend to his appearance.
A warm breeze came through the open window, gently ruffling Sophie’s curls and pulling at the piles of written work, so neatly laid out on the table’s surface. Dunbeath moved his head closer to Sophie’s as they pored over a mass of celestial charts and interplanetary readings.
She now tapped with her finger at a table covering the measurement of parallax angles.
‘I have thought about our problem overnight,’ she said, ‘and I think this is where the error has been stemming from. You see, these lunar distances are changing over time at a rate of half a degree, or thirty arc minutes every hour …’ she continued to take Dunbeath through her detailed reasoning and a few minutes later approached her conclusion, ‘…thus I think this could be inducing an error of as much as one quarter degree in longitude or about fifteen nautical miles at the equator. The core issue is one of timekeeping and I’m sure that by always using the periods of the Jovian moons rather than working with a clock we can be sure to resolve this.’
Dunbeath sat back. Sophie continued to study her workings but out of the corner of her eye she was conscious that Dunbeath was looking longingly at her from the side.
‘I knew I had gone astray somewhere, Sophie,’ he said at last. ‘I knew the perfection of the heavens couldn’t be at fault. It had to be my error and thanks to you it’s now exposed – and furthermore you have a solution. I can scarcely credit this, indeed, I can scarcely credit everything that has happened recently. At last all my years of work and research have come together. That it has is much due to your help. And, of course, to your observation of the Transit.’
Sophie began to bridle gently at his thanks but Dunbeath wouldn’t be deflected.
‘I’d worked for twelve years without missing a single reading,’ he went on, now looking closely at her, ‘and yet you saved me when I could not continue. And then you’ve exposed these great errors of mine. Now, at last, the answer is here. You must know how much I am in your debt, Sophie, and I believe these findings are now in a position to be presented at the next meeting of the Board of Longitude. It should be held soon – there’s usually one sometime in the spring. They’ve often been in May in the past and, in fact, I’m surprised I haven’t been told of a date yet. I’m convinced these findings will put me so far ahead in the race to satisfy the criteria for the measurement of longitude that the Board will be hard pressed not to award me the Prize immediately.’
He looked again at Sophie and his tone changed. He began to speak to her in a voice of quiet intensity.
‘Sophie, I couldn’t have achieved this without you. You may have saved my life when you nursed me through my illness but you saved my life’s work too. Will you agree to come to London with me when I go to the Board? I want them all to know of your contribution.’
Then he took a deep breath as if preparing himself to say something vital.
‘Sweet Sophie, I want you to be with me then – and I want you to be with me when we return. Nothing must come between us. You’ve brought this old castle back to life. More than that, you’ve breathed a new life into me – you have changed me. I want you to stay here. Will you consider this?’
Sophie reeled in panic. What was this? Was it a prelude to an offer of marriage? Or did Dunbeath just see her as a collaborator in his research? She had thought she understood him for the impatient, distant man he was …and yet here he was calling her ‘sweet Sophie’. That was singing love songs from the hills for a man like him. She may only recently have washed up in Scotland but she knew enough about these aristocrats to know that they saw marriage more as an opportunity to add wealth and power than anything to do with the heart. And Hume had told her that gain was all that motivated the Urquhain. She hadn’t anything to offer. So what could he mean? She thought further and then firmly decided not to press him – she might not wish to hear the answer. She just had to buy time. There were still thirty nine days before her father’s debt fell away – and she had to have Dunbeath’s protection until then. Whatever it was that he meant, she had to play for time.
Sophie gazed at Dunbeath, looking deeply into his eyes. Then she blinked and looked away.
‘Well, my lord, I’m flattered by what you say. Indeed I am greatly affected. I owe you so much too. You forget that. You, too, have saved me.’
As she spoke she listened to her heart. There was undoubtedly a great affection there for the man, in spite of his anger and restless arrogance. But perhaps what had begun as a fascination in her had not yet turned to love? Perhaps it would? And did it matter anyway? If he should mean to press for marriage then the security and status that he could give her were far more than she could ever have hoped for. In any event, marriage or not, if she stayed with Dunbeath she would be able to do so much – she would greatly prize working with him to continue their research together – and there was little doubt that his great wealth could help her poor father as well.
In an instant she had weighed up her position: her heart might not be full but that could change in time, particularly if his was truly full for her. And her head told her not to show any sign of rejection. She had few options anyway, and anything that protected her from Zweig and the debt had to be agreed to.
‘Yes, I would be very happy to go to London with you,’ she said warmly, carefully ignoring any deeper message he may have meant. ‘Thank you. I greatly look forward to hearing of the date.’
She looked down as she tried to steer the conversation away from any further discussion about their future. Her eyes were now on the table and she stretched out a hand to pick up the Domenico Salva from where it lay on a sheet of planetary movements.
‘This really is the most beautiful piece of work,’ she said, looking at the chasing of the gold case. She ran her finger over the jewelled exterior and then lifted the telescope to her eye and focused it on a boulder set in the beach at the far end of Dunbeaton Bay.
‘And what superb lenses it has. I’ve never seen another like it.’
She set it down again and continued to speak quietly and with no small trepidation. She knew the time had come – she had to find out what Dunbeath knew of Zweig.
‘But, why do you think that ship’s captain had it? Did he steal it from you? Do you remember he said - ‘how do you think I came by it?’ - when he jumped out at us on the dunes? What do you think he meant by that?’
‘I don’t know,’ replied Dunbeath, suddenly his old sharp self again, ‘I’d never seen him before.’
‘Yes,’ Sophie carried on, a slight trace of incomprehension entering her tone, ‘but what do you think he meant?’
‘How could I know what he meant?’ Dunbeath was beginning to bridle. ‘I told you, I’d never seen him before. I had no idea who he was.’
* * *
Once Zweig had returned the wooden fake to the original’s hiding place he’d turned his thoughts towards how he could steer James away from his growing obsession with having the telescope returned to him. The boy had been becoming more querulous by the day and he was now insisting on seeing it again. The previous afternoon, he had even spoken of exposing Zweig to the English army if he didn’t get his way. At least with the fake to draw on, the captain felt, he had something that he could wave in front of him if the need arose. In spite of this, he knew that James could only be held off for so long and Zweig now decided that he had to bring forward the second part of his strategy.
The boy was a factor to be considered, that was true, but he was just an irritation compared to the larger plan to win Sophie round. Zweig knew her mind, though, and he was quite sure that the debt schedule was the leverage he needed for his next steps. Sophie had already shown him how aware she was of the dates – and she would know only too well that time was running out for them to return to Königsberg and so trigger the repayment. He could be certain that she would see this as the ticking clock. A further two weeks had passed since he’d seen her on the dunes and he could be sure she would be aware that there were now just thirty nine days left. Many fewer and she would think that Zweig wouldn’t be able to make the journey to be back within the hundred days that Kant had set. He couldn’t leave things any longer – now was the time to bring matters to a head.
He took a deep breath. He knew there would be much pain in the next part of his strategy and he set his shoulders as he steeled himself for the ordeal ahead. After a few moments he shook himself slightly and began the short walk to the castle. It wasn’t long before he came to the top of a large dune and he stopped there to look down at the granite slab that made up the side of the great Urquhain stronghold.
About a hundred yards from the vast fortress the constant working of the tides had so eroded the front of the dunes that a small raised peninsular of grass and sand stretched out onto the beach. The end of this was in full view of every window on the castle’s north face and it was here, he had decided, that would be his battleground.
Having satisfied himself again on his choice, Zweig walked towards the hummock. By now there was such evident purpose in his stride and such great determination in his features that it was plain that his mind was quite set.
He reached the end of the spit and turned towards the castle, checking the sight lines from the windows. Apparently satisfied that nothing would obstruct a view of his position he sat down and carefully folded his legs under him in the way he’d seen practiced by the monks of the east.
Then he closed his eyes.
Above him the sky darkened as rain clouds blew in from the sea. Now, facing towards the largest windows, his eyes still shut, his body stiffened as if he was summoning up his inner strength.
Then he opened them again and his face was composed in a trance of concentration. He stared fixedly at the castle and he tensed again as if distilling every particle of energy to serve his colossal will. Then he leant forward from the waist with such an intense show of force that some strange power seemed almost to shimmer off his huge frame.
This will take much time, he thought to himself. He had to be patient, he had to stay alert.
* * *
Hume and Dunbeath had moved their game playing from the salon to the dining room. For much of the last two weeks Dunbeath had been taking part only intermittently, his mind more occupied by the navigation problems that Sophie had recently resolved. Hume had not concerned himself too greatly with this delay and had instead attacked the great Urquhain library, devouring whole sections of the Greek thinkers that he’d found on its shelves.
But, with the solving of the celestial problems, the Dilemma was drawing the two men together again and they now sat opposite each other at one end of the long table, the record of their recent turns spread out between them. Flames from half a dozen enormous logs in the great fireplace threw their shadows about the room and the ornate plasterwork and fierce staring caryatids seemed almost alive in the glow. Above the pair’s hunched backs, a vast barrel shaped ceiling of raised strapwork and long, pendulous roof bosses loomed over them, a masterpiece that dated from the time of King James.
Sophie sat next to Dunbeath, glancing occasionally at a large sheet of paper on which she’d been keeping the score.
Hume sighed and sat back in his chair.
‘We do not seem to be progressing at all, Dunbeath. Yet again you have chosen to defect. Whilst I know your logic in doing this I can only repeat that I can see no encouragement to deal with someone who never shows any sign of reciprocating my evident wish to co-operate.’
‘But Hume,’ replied Dunbeath crisply, although in a far less provocative tone than he’d used during their earlier games, ‘even though you know how I shall always play I still don’t understand why you can’t see that I am beating you. And that must surely be the mathematical lesson of the game.’
‘No,’ said Hume firmly. ‘Sophie and I were discussing this yesterday and we’ve both concluded that it is exactly this belief that exposes the Dilemma’s fundamental principle. And it is this. We believe it is a mistake to talk of ‘beating me’ or to see the game as one of winners and losers.
‘If this is so, then nothing is ever added, nothing is ever built. Sophie has called this a ‘nil sum game’ because if one man’s gain is exactly matched by the other man’s loss then there is no gain, nothing is created, the sum total is, indeed, nil. But even the small boys with their cake showed that co-operation – although it was based on greed and the fear of losing –can add something. In their case the gain may simply have been an outcome of unusual tranquillity but if one thinks of it as peace between warring factions then that is indeed a giant step forward.
‘The trouble with these nil sum games is that if one man is winning only at the expense of the other person losing then there will never be a lasting relationship. How can you ever trust someone who you know is always trying to beat you? Indeed, did we not come near to losing our friendship over your own strategy? And was it not your insistence that you were beating me and that this was all that mattered that led me to give up on you? On the other hand any merchant will tell you that while he would wish to profit from you, he wishes you to profit also. In that way you can both be satisfied. And the likelihood is that the two of you will then continue to trade.’
Hume stopped for a moment and looked into the fire. He had given the Dilemma a deal of thought these past two weeks and he now wished to bring the sterility of their repetitive contests to an end.
‘Let me give you another example,’ he continued looking back to Dunbeath, ‘you’ll recall that in our grandfathers’ day the English and the Dutch spent ruinous sums on great navies to protect their commercial interests as if the gain of one would be the loss of the other. But we learnt, did we not, that it was actually due to the competition between them that world trade grew. It became evident that it was not the share of the trade that one country achieved that mattered so much as the growth in the custom itself. Both countries could be winners – in fact, the evid
ence was that they were.’
‘You may be right in both your examples,’ Dunbeath replied thoughtfully, ‘but nonetheless it is the rational player in these games – as in life itself – who will do better when he plays an irrational one. These leaps of logic you describe may be obvious to the distinguished moral philosopher I am addressing here at my table but they will escape the normal man. They are therefore not present in everyday exchanges. Far from it, it is the calculating fellow who will always beat the naïve.’
Hume sighed heavily and his shoulders sagged.
‘Sophie, can you help me here? I really don’t think this is a conclusion that I alone have reached. I think we only have to look at how people go about their daily lives to see that.’
Sophie looked down at the scores.
‘Perhaps it’s easier for me to give an opinion as I haven’t been caught up in the competition of the game,’ she replied, ‘but it must be plain that the answer about how to play depends on what one’s aim is. If one wants to gain from society by building the order and reliability of that society, and therefore to profit from it, then co-operation must be the correct strategy. Essentially the way I see it is that, over time, one is not escaping the hangman’s noose of a one-time Prisoner’s Dilemma but instead looking for people to play high scoring games with – to live to fight another day rather than to be beaten. To win wars not battles. And to do this one has to recognise that it’s the collective score that matters more than the individual’s, just as keeping silent would have helped both prisoners if they could but have trusted each other.’
Hume nodded at this and smiled encouragingly at her as she continued.
‘Unlike the prisoners and your attempt to replicate the two cells with your book barrier – in real life we do communicate. In reality we remember how people have behaved in the past and how they are likely to behave in the future. And therefore we can decide who we want to deal with. On the other hand, the key to dealing with constant defectors is to decide not to play with them. To avoid them and find other partners instead. Once one’s found co-operative people then stable relationships and stable societies can be formed that will bring real benefits to the individual. That is how a man wins, not by constantly trying to beat someone in every single exchange. This much seems evident to me. However, it also seems that the real problem then arises with people who take advantage of a society in which others are co-operating.’
The Prisoner's Dilemma Page 16
