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Escalation and what it means for Trident

Campaigning for the general election is nearly done, but one curiously high profile issue has been nuclear weaponry. Find out how a simple mathematical model can show how all the leading parties are probably wrong.

Despite the fledgling  economic recovery, a health service with lots of challenges and a growing divide between rich and poor, nuclear weapons have been a distinctive topic in this election. Of course one of the reasons why it has been sensitive is because of the prioritisation of renewing our deterrent at a time of so called austerity. Nuclear weapons are a prime example of how pure mathematical thinking can be applied to real world problems. Here we’ll demonstrate a simple model and use it to show that all of the leading parties in this election are probably some way off.

The cold war was an arms race like no other, and hopefully we’ll never see the likes of it again. But countries take their security very seriously and we are still living with the legacy of massive weapon accumulation. As the UK considers whether to keep its Trident nuclear weapon system, we can consider why we have it in the first place and how we can reduce the burden on the public finances in the future.

A simple model of the cold war is the dollar auction, which helps explain why countries continued to accumulate so many weapons to a wholly unnecessary level. We’ll consider the simple case where there are just two players.The prize for winning the auction is a dollar. On each turn each person has the option to bid more than the other player’s bid. But if they do not bid, they lose the money they have previously bid. If they win they only get the dollar, and they don’t get their bid back.

Player one goes first and its a no-brainer. Either he doesn’t bid and doesn’t get the dollar, or he can bid just 1 cent for a chance to win a dollar. Most people are willing to risk a cent for the chance to have a dollar. Player two now has a similar choice. Not quite has good as he must bid 2 cents, but hey, that’s a small risk for a whole dollar. Play can continue like this for some time.

But there comes a point where the players objectives change. Consider the situation where the current bid is 96 cents from Player 2. Player 1 has the choice of losing his 95 cent previous bid, or making a 97 cent bid and making a small 3 cent profit. Of course, a profit, however small, is preferable to a large loss, and hence another bid is likely. The same is true when the current bid is 99 cents – another bid will reduce the profit to break even, but the alternative is to lose nearly a dollar. So the bid is increased to a dollar.

Back to player 1, who had a choice of losing his previous bid of 99 cents, or increasing his bid to $1.01. If he doesn’t bid he loses 99 cents. Or he can bid and lose just 1 cent. So he now bids, not to win, but to minimise his losses. And so it continues. But the players are now bidding more than the prize fund was worth.

Now the objective is to minimise losses, and play can continue indefinitely. Losing a dollar is still better than losing two! But the game really goes to extremes is prestige becomes an issue. There is little difference between losing $99 and $100, except that players have now invested so much they are concerned about the loss of prestige and criticism if they did lose. The final change in objective is to make the other person lose, not to win.

The link to an arms races should now be clear. Like the dollar auction, nations only need to spend a bit more each year in their defence budgets to stay ahead, but the long term net effect is that they spend way more than the initial objective would warrant, despite making seemingly rational choices at each step. It explains why the US and former Soviet Union went so far as to accumulate a nuclear arsenal that could flatten the world several times over, when once would surely be more than sufficient!

Of course you may have noticed one obvious flaw with this game. At each stage we have only considered that a player has only two choices – bid or lose. It is clearly easy to look ahead and forsee the bids increasing. If the players wise up they could see where this is going and might stop bidding earlier. But the same can be said of an arms race. Had the US and USSR had the sense to see where things were going in the post war period, it might never had got so serious as it had by 1980.

Furthermore, studies have shown that individuals are likely to under estimate the likelihood that the other player will bid again, or underestimate the amount the other is willing to bid. Most people’s reaction to an outline of the rules of the game is that you shouldn’t bid more than a dollar!

So how does this relate to the current debate about Trident? Renewing Trident is effectively perpetuating the dollar auction – upgrading the existing system is equivalent to making another bid.

Abandoning Trident altogether is equivalent to losing the auction, wasting everything that has already been invested. This is a concern when others (Russia) have clearly express their intention to continue the game.

The best way out of a dollar auction is to agree to unwind the auction step by step. It allows players to reduce their losses step by step, and check that the other follows suit. In that respect the Liberal Democrats policy of reducing Trident by 25% is probably closest to the ideal, but it lacks the essential ingredient of agreement with other players.

So it would seem that the ideal policy choice has not been adopted by any of the leading parties in our UK election debate.

Of course there are plenty of other reasons to retain nuclear weapons. For example the ongoing issues with rogue states. Also a far more likely scenario for the use of nuclear weapons is tactical, to resolve a bloody stalemate for instance. Imaging being able to tell a military commander in 1916 that they had either the option of another 2 years of stalemate, or they could blast a hole in German defences and bring about a swift conclusion with net fewer casualties. Indeed many plans were pursued during the war to do just that.

But the purpose of this article is not a complete exposition of the pros and cons of nuclear warfare. Rather it was to show how a relatively simply model can be applied to a very real life scenario and offer useful insights and inform policy choices. Of course there are more complex models which offer more information, but these could not be explained in the course of this already long article. The dollar auction is now used as an explainer in a wide variety of fields to explain how escalation can cause a conflict to continue beyond points which would otherwise seem rational, even when the players are trying hard themselves to be rational. 

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