The Game Theory Behind Golden Balls

Golden Balls is a simple, fascinating and sometimes hilarious game show in the UK that must have been created by an economist. It is a game show where prizes should theoretically never be won. Yet to the dismay of the economist who orchestrated the game something very interesting happens in reality.

The game involves players accumulating prize money in two rounds of pre-play. In the final round players have the opportunity to share or steal the accumulated prize money. This round is a one-stage game where two players can communicate briefly before simultaneously choosing whether to “Split” or “Steal”. If both players Split, the prize money is distributed evenly. If one Splits and the other Steals, the latter wins all the money. Finally, both Splitting means no prize money is won at all. The game can be written in normal form below, where M represents the accumulated prize money, green relates to the first player and orange to the second:

P1\P2	Split	Steal
Split	M/2 , M/2	0 , M
Steal	M , 0	0 , 0

This game is a “weak prisoner’s dilemma.” For the economists reading, it has three pure-strategy Nash equilibiria in (Split, Steal), (Steal, Split) & (Steal, Steal), as well as an infinite number of mixed-strategy equilibiria where one person Steals with certainty and the other randomises. For those that have not studied game theory, the analysis of the game is simple. If your opponent chooses to Split, you should choose to Steal since winning all the money is more than half of it. If your opponent chooses to Steal, you are indifferent between Splitting and Stealing. Given that Nash Equilibrium relies on knowing what the other player is going to play, it seems most rational to always play Steal. In other words, if there is a non-zero probability of your opponent ever playing Split, then you should always play Steal. This is because Steal is a weakly dominant strategy (you are never worse off by playing it). Only if you correctly believe your opponent is choosing to Steal with certainty can a (Split, Steal) equilibrium be possible.

It is called a prisoner’s dilemma because the most likely equilibrium of mutual Stealing is Pareto dominated by mutual Splitting. In other words, both players can be made strictly better off if they deviate together to Split. However, in a stage game without repetition, this equilibrium should never occur. If you believe your opponent will Split, you should always Steal. A working paper at the University of Zuirch* found some very interesting results. By studying outcomes of the game it finds that there is a 33% mutual cooperation rate of both Splitting. The paper then goes on to investigate what increases the likelihood of both players Splitting such as handshakes, racial bias and so on. However I am more interested in why Splitting could ever possibly result in equilibrium in a one-stage game such as this. I came up with two potential explanations, one of which I dislike but felt the need to mention and the other which, with its pitfalls, does seem to explain the data.

1) Miscoordination

It is important to realise, given the payoff structure, that both Splitting is not a matter of mutual cooperation but instead a situation of miscoordination. The reason being is that without repetition of the game, there is no incentive to coordinate. Once one player Splits, the other must Steal. Hence the cheap talk before choosing your action is meaningless. If you have convinced your opponent to irrationally Split then you should rationally choose to Steal. As a side note, I find it ironic that players try to convince each other to Split with the lure of Splitting as well. Theoretically the best strategy to ensure you win the entire prize is to convince her you will Steal. Only in this situation will she be indifferent between her two actions. When she believes you will Split, she will be better off by Stealing.

The reason both Splitting can be a result of miscoordination is when both players choose different Nash equilibrium. If one believes that the equilibrium will be (Split, Steal) and the other (Steal, Split) then both Splitting may result. In other words, if both players believe their opponent is Stealing with certainty, an equilibrium where they both Split is possible. The problem with this explanation is that Nash equilibrium relies on correctly knowing your other player’s strategy. As I mentioned above, if there is any non-zero probability of a player choosing to Split then the opponent should always choose Steal. Hence, you would only really choose to Split if you are 100% sure your opponent is Stealing and hence you are indifferent. 

Furthermore, your payoff from mutual Stealing may actually be higher than when you Split and your opponent Steals. This is because in the former you “fail to win” but in the latter you have “lost.” Or put more simply, if you know your opponent is going to Steal, are you really indifferent between Stealing and Splitting? Perhaps I am filled with too much animosity but I would rather my opponent got nothing than stole everything. Hence in this sense, stealing is a strictly dominant strategy and miscoordination cannot be a good explanation.

2) Payoff Re-model

The results of the Zurich study are interesting. Stake size and communication do actually have significant effects on the outcome. There is a negative correlation between cooperation and stake size. Actions such as mutual promising to Split and a handshake increase cooperation. This suggests that the payoff of players do not wholly depend on the monetary gain. I believe we can crudely remodel the game with an additive term to the Split payoff of k(x), where k is increasing in x and x is a set of characteristics such a reputational concerns. They key point is that x can be endogenously given, in the sense that the cheap talk and affection towards your opponent can affect it. For example, if you do not want to be seen stealing from an old lady who you really admire on national television then x may be very high. The normal form becomes:

P1\P2	Split	Steal
Split	M/2 + k(x) , M/2 + k(x)	0 + k(x) , M
Steal	M , 0 + k(x)	0 , 0

Now Splitting is a dominant strategy for a particular player if their value of k(x) is larger than M/2, meaning that regardless of what the other player does, this player should Split. If this is the case for both players then (Split, Split) is a dominant strategy equilibrium. For example, a handshake and a mutual promise will greatly increase reputational concerns or guilt from not Splitting and hence lead to a large k(x) for both players. Note we could have modelled the game by subtracting k(x) from the Steal payoffs as well.

The larger the monetary prize for a given level of x, the less likely k(x) will exceed M/2, explaining the negative correlation between prize money and cooperation. In other words, at some prize level, the potential gain from Stealing all the money exceeds the costs of a damaged reputation or guilt.

Where one player's k(x) exceeds M/2 but the other player's does not, the only Nash equilibrium is (Split, Steal). Where neither player's k(x) exceeds M/2, both (Split, Steal) and (Steal, Split) are Nash equilibria. These explain the unilateral cooperation rate of 55% by players.

The problem with this revised model is that now (Steal,Steal) isn't ever a Nash equilibrium if k(x) is some non-zero value. In light of this, it might be more appropriate to remove k(x) when the other play Steals meaning that players only avoid repetitional damage by Splitting when the opponent is Splitting as well.

Another problem with this argument is that it simply restates the payoffs rather than provide rational explanations of cooperation in the original game.  However, if players are only concerned with their monetary payoff, then how can you explain a situation where both Split?

* http://www.econ.uzh.ch/faculty/graetz/publications/wp1006.pdf

The Tale of the Taper

It is apparent now more than ever just how much US monetary policy affects the rest of the world. The Fed has been printing $85bn a month, half of which buys government treasuries with the rest going towards mortgage-backed securities. If you read my last blog, you will understand that this unconventional monetary policy is called quantitative easing (QE). Whereas most central banks have adopted QE, none have quite had the same far-reaching effects that the Fed’s QE3 has had on the global economy.  A minor change in US policy can lead to global catastrophic effects.

In May, Ben Bernanke, the head of the Federal Reserve, outlined that they would begin to “taper” their QE3 programme in the coming months in response to positive US economic data. Little did he know the ramifications that statement would have. Tapering involves reducing the amount of money that the Fed will print each month, not to be confused with stopping the policy altogether. In fact, following his remarks, the markets predicted that he would reduce QE from $85bn to around $70bn by September’s meeting. This widely accepted expectation lead to huge knock on effects.

Most obviously, bond yields soared as expectations of future interest rates increased. To understand this, you must know that bond yields move inversely to their price. If the Fed were to reduce demand for US treasuries then their price would fall and subsequently yields would rise. The expectation of this and higher future interest rates pushed up yields on longer-term products. In fact 10-year government bonds jumped a full percentage point following Bernanke’s revelation, and not just in the US.  As yields rose, bond prices fell hurting those with a large portfolio of fixed income products, wiping away large amounts of capital.

Then there was the "Emerging Market Summer Sell-Off". The anticipation of less liquidity due to tapering and improved returns in the developed world, lead investors to begin to pull out capital from emerging markets. The hardest hit were India, Indonesia and Turkey. Loose monetary policy in the US lead to huge capital inflows into these economies over the past few years. As investors began to pull out of these emerging markets, their currencies began to tumble. With large current account deficits, the central banks had to start eating into their reserves to prop up the currency. In fact, Raghuram Rajan, the newly elected head of India’s central bank, decided to increase interest rates last week to protect its currency. These economies rely heavily on huge capital inflows to fund their deficits, so the news of a potential tapering was a disaster.

There have been a number of other consequences. The stock market has been volatile. Bulls will say that tapering indicates the US economy is on a upward growth trend whereas Bears will argue that any monetary tightening is too early and would upset equities. As a result, risk appetite has fluctuated according to investor confidence and the expectation of when the tapering will begin. That said, the news of a taper has not prevented a surge of M&A activity, including the third largest acqusition ever between Vodafone and Verizon Wireless for $130bn.

Gold has also fallen dramatically over the last few months as people move into riskier assets and thanks to an expectation of a dollar increase (due to less tapering).

The most important thing to note is that all this happened before tapering had even started. Everyone was pricing in a taper before it had happened. So it surprised the majority of the financial world this September when Bernanke decided not to taper.

Even though US unemployment had fallen to 7.3%, closing in on Bernanke’s 7% target of completing the taper entirely and the 6.5% US target of increasing the Fed Funds rate, the Fed decided to stick with $85bn per month. The explanation was three-fold even though it was made clear that it was a ‘borderline’ decision. Firstly the reaction in and of itself was enough to suggest that the US economy was not ready for a taper. Secondly, the data was not as convincing as first glances suggest. Although unemployment had fallen in the US, this is mainly thanks to a disastrously low participation rate; it’s lowest for decades. Lastly, the US still needs to sort out the mess with its debt ceiling. If Congress does not reach an agreement on increasing the debt level, the government may have to shut down all non-essential activity in October. Although very unlikely, Bernanke was implicitly keeping monetary policy loose in case of a severe fiscal tightening in the coming months.

So was he right? Yes and No.

It certainly benefited the emerging economies. The news of no taper stemmed the flow of capital flying out of emerging markets and helped bolster their currencies. They now have a window of time to correct the imbalances in their economies before the inevitable repeats itself. Furthermore, a potential taper spooked markets and Bernanke is correct that there is no need to rush monetary policy tightening, especially when inflation lies below 2%. It would be a shame to damage the long-awaited return to growth with a premature increase in interest rates. Lastly, the housing market is at the centre of the US growth trend. An increase in rates will only damage this growth. Following the announcement, the S&P 500 shot up suggesting investors were delighted with the decision to keep money ultra-cheap.

However, the flipside is that everything had already been priced in. Bond yields had shot up, the stock market was positioned ready for a taper and currencies had moved accordingly. There is a strong argument to say that the consequences of a taper had already been felt and the actual tapering was just a formality. Furthermore, its important to realise that tapering is not monetary tightening. Reducing $85bn to $70bn is simply less monetary loosening. So given that the US is now growing and unemployment falling, it certainly seemed the time to slow the stimulus down even slightly.

However, the most important factor for me is central bank trust.  A key tool for a central bank is being able to communicate with markets. Markets must trust central bankers and believe their promises. It is now very unlikely that QE3 will be fully wound up by the time unemployment hits 7%. This makes the central banker’s communications far less credible. If they never stick to their plan, what’s the point of listening in the first place?

Furthermore, the early stock market gains were quickly subdued as investors began once again to argue over when tapering will actually happen. The fact that the Fed decided not to taper also dampened investor confidence in the US growth trajectory pushing a lot of people back into safer assets such as government bonds. In effect, good news quickly became bad news.

So markets are still left volatile and the process begins again for the October meeting. The new expectation is that tapering will begin in December but the herd was wrong last time. I believe that the markets were prepared for a September taper, which had already been priced in. The only thing not tapering has done is attach a large amount of unpredictability to the central bank.

A New Era of Monetary Policy

Over the last few years, monetary policy has changed dramatically. Newspapers are littered with terms such as interest rates, money supply and quantitative easing. If you are new to Economics, and interested by what all this means, this may be a good article for you.

This blog explains the basics behind monetary policy before exploring how it has evolved over recent years. I finish with evaluating “forward guidance”, a new monetary policy introduced by Mark Carney, the governor of the Bank of England (BoE).

What is an interest rate?

An interest rate is simply the cost of money (or the price of borrowing money), where ‘money’ is literally the banknotes in your wallet or the balance in your checking account. There are obviously many interest rates, however, when a central bank sets the interest rate, it is referring to the (risk-free) short-term rate on cash deposits, such as the overnight rate between banks.

The most basic economic concept is that in all markets, the price of a good is the equilibrium between supply and demand. An excess demand over supply will lead to a price rise until demand equals supply again and the opposite is true for an excess supply. The money market is no different. People and firms demand money for transactions and the central bank supplies it. Since the interest rate is the price of money, it sits where money demand is equal to money supply. If money supply outstrips money demand, the interest rate will fall until a new equilibrium is reached. Therefore when a central bank says they are reducing an interest rate, what they are really doing, behind the scenes, is increasing the supply of money. By injecting more money into the system, the price of money (i.e. the interest rate) falls.

Conventional Monetary Policy

Prior to the recession, conventional monetary policy involved the BoE simply setting the interest rate (the base rate) each month. If the bank wanted to reduce the rate (for example down to its current level of 0.5%), it would do so by increasing money supply. The BoE does this by buying safe, short-term assets, usually government bonds, from the market. It pays for these assets with money hence increasing the amount of money in the system (increasing liquidity) and reducing the interest rate, as explained above.

The reason the BoE reduced the interest rate so dramatically was to boost the UK economy. The idea is that a reduction in interest rates would encourage spending over saving. A consumer’s return to saving (and the opportunity cost of holding money) falls. Consumer spending is the largest part of GDP and hence the most important factor in boosting an economy. Lower interest rates also encourage a to firm invest more since the cost of financing that investment has fallen. Furthermore, the demand for the pound falls as interest rates fall, depreciating the currency. This in turn increases demand for imports and decreases demand for exports, improving the trade balance. All these factors contribute towards growth.

With any policy come the unintended consequences, which in this case, is inflation. As aggregate demand increases so does the general price level. In fact, the BoE have an inflation target of 2% and must do the very opposite (increase the interest rate) if inflation is too high.  High inflation can be dangerous for an economy for several reasons.

More importantly, excessive monetary policy can lead to a ‘liquidity trap’. This occurs when increasing the money supply can no longer reduce the interest rate. When an economy’s interest rate is close to zero, a central bank can no longer reduce it. Most economies found themselves in this position during the financial crisis, which made them turn to other unconventional methods of boosting the economy, such as quantitative easing.

Quantitative Easing

Many economies turned to QE when rates could be lowered no more. On a very basic level, this involves the central bank buying longer-term and therefore riskier assets, such as longer-term government bonds (gilts). They do this simply by printing money. Technically, this is not true. Central banks use banks as intermediaries. In effect, banks buy longer-term assets from financial institutions, which they in turn sell to the central bank in return for a higher reserve balance (a bank account that a bank holds with the BoE!)

The whole point of this is to reduce longer-term interest rates (‘further out on the yield curve’) and increase the supply of money further.  Conventional monetary policy focused only on short-term interest rates. Although lower short-term rates can feed through to longer-term rates, this is a more direct way of doing so. 

Since yields on government bonds have fallen, is it cheaper for companies to raise capital - lower longer-term interest rates encourage even more investment. As long-term returns to fixed income products fall, investors turn to the stock market instead for better returns, boosting share prices. Most importantly, the hope is that those firms that have sold their assets, spend the money they receive. As banks increase their reserve balances, the hope is that they also will lend more, leading to cheaper long term borrowing costs for the consumer. In the UK, the asset purchase facility (QE) stands at £375bn.

The problem with QE is that if there is no spare capacity to absorb the increase in spending, the policy can be dangerously inflationary. Printing money can lead to hyperinflation such as in Germany in the 1920s post WW1.

Furthermore banks may not lend out their additional reserves (cash hoarding) and institutions may sit on the extra cash without re-investing. There is no policy to force these institutions to spend the increase in money supply.

Forward Guidance

Mark Carney, the new governor of the BoE, was made famous for forward guidance when he implemented it in Canada. He has now done so in the UK. The idea is simple promise. Carney has promised the UK that he will not increase interest rates until unemployment falls below 7% (currently at 7.8%). This has very similar effects to QE. It flattens the yield curve, or in other words, reduces longer-term interest rates and expectations of future interest rates. Carney said he does not expect the interest rate to rise until at least 2016.

Forward guidance is supposed to provide the markets with reassurance and confidence. It allows firms to invest knowing they will not be hit with higher costs of financing their debt in the medium-term and encourages consumers to buy mortgages with the knowledge rates will stay low for a while yet. In summary, it increases spending and borrowing.

Yet the FTSE tumbled and the opposite happened when forward guidance was announced. The first reason for this was that Carney anchored an interest rate change to another variable – unemployment. Tying a future rate rise to the level of unemployment leads to a lot of uncertainty. Firms and consumers cannot accurately predict when unemployment will fall to 7%. In fact, the markets believe that the BoE is far too pessimistic and are predicting a rate rise much sooner in 2015. It also makes a mockery of the monthly unemployment statistics as firms and net borrowers sit and hope that the jobless remain jobless so that rates do not rise. The more optimistic view of the markets hence meant that the yield curve was not flattened as much as Carney would have liked.

Furthermore, there are too many get-out clauses that make the date of a rate rise far too unpredictable. If medium-term inflation expectations become too high then rates will rise. As we have seen, low interest rates cause inflation. With inflation currently at 2.9%, markets fear that this get-out clause is a possibility. Furthermore, the Financial Policy Committee (FPC) can force a rate rise if they believe the policy causes “potential systematic risks”. In other words, the rate can be raised at any time.

My conclusion on forward guidance therefore is that it is a great idea if it is bold. If you can make a credible commitment to keep rates low for a certain amount of time, like he did in Canada, then forward guidance can have the intended consequences. Albeit very risky, a promise such as “I will not raise the interest rate until 2016,” would certainly have the desired effects if credible. However, by tying it to the unemployment statistics and providing too many get-out clauses, the policy does not do much at all. In fact, all has done is bring forward the markets’ expectation of when rates will rise.

Immigration – Dispelling the Myths & Ignoring the Ignorance

Humans have an innate desire to place blame and find a scapegoat. It therefore does not surprise me that immigration in the UK is a touchy subject when it absolutely should not be. The UK, now more than ever, is in dire need of a large influx of immigration yet David Cameron has succumbed to the irrational public pressure to reduce it. This is a perfect example of politics interfering with sound economics.

The conservative party wants to limit net immigration to 100,000 people per year. It has already tightened the rules and imposed a cap on student visas and set a limit for non-EU skilled immigration. Luckily nothing can be done about EU immigrants who have a right to migrate, but even this, which accounts for a third of net migration, may come under scrutiny following a future EU referendum.

Let’s be clear. Immigration in a globally competitive world is fundamentally good and in the case of the UK, absolutely necessary. UK net immigration needs to increase not decrease. The Office for Budget Responsibility (OBR) has been clear on this. UK demographics are changing and our population is ageing. In order to prevent the UK from going bankrupt, we require a large influx of immigrants to pay off our overwhelming debt burden through taxes and work.

Usually anti-immigration arguments are riddled with myths, ignorance and xenophobia. It is time to dispel the myths of immigration so we can overcome the ignorance.

Dispelling the Myths

“Immigrants cost us money”

Perhaps the most common and factually incorrect argument is that immigrants cost the UK taxpayer money. The anti-immigration clan fall back on the idea that many immigrants sponge off the state and use the UK as a health or benefit tourist destination. This is wholly untrue and to some extent simply racist propaganda.

Of course there are the few that abuse the system just like there are a minority of UK citizens that do so as well. Yet the overwhelming majority of immigrants are of working age and hold jobs. They pay taxes, national insurance and contribute to spending and our GDP. Most are skilled and many hold university degrees or equivalents. Immigrants add to the economy, increase productivity, increase the flow of ideas and innovation, improve trade links and add new specialisations to a rather one-dimensional economy. Most importantly, as the OECD explains, immigrants without a doubt put more into the pot than they take out.

It is for these reasons that the OBR has recently published a report saying that the UK requires an influx of immigrants to sustain public finances and counteract a talent shortage. The UK is an ageing population. The OBR says that a net of 140,000 migrants per year from 2016 would lead to a net debt to GDP ratio of 99% by 2062. With zero net migration, however, it will rise to an obscene 174%. To maintain the UK 2.75% long-term trend growth, we need the same immigration levels as when Britain was opened up to Poland. Sir Alan Budd calculates that due to changing demographics, trend growth will naturally fall to 2% (an almost 30% decrease in the long run growth rate), unless immigration increases.

State pensions, social care and healthcare will rise from 14% of GDP to almost a fifth in the near future.  Immigrants can pay for this. Hard-working, productive, tax-paying immigrants, usually in jobs that they are far too qualified to be in can fill the gap. If the Tories continue with their plans, a lack of immigration will completely counteract years of austerity and put the UK public finances back in the same dire position.

Of course, a larger population of immigrants leads to larger NHS and welfare costs. The point is not to focus on only the costs when the benefits so obviously outweigh them. Immigration pays for itself and so much more. Many argue that immigrants from the EU, who are free to move, abuse our health and benefits system. But this is also simply not true for the majority. EU immigrants are, on average, younger and more likely to be in work than a native. Ironically, Spain suffers from elderly Brits spending their later years on its beaches.

 A lot of the skepticism towards immigration is simple scapegoating. Something that I hope will one day be eradicated by globalistion and subsequent acceptance of other races. Immigration does not cause recessions. Recessions are caused by bubbles, greed, stupidity, risk taking and poor judgments.  In fact, immigrants can smooth the costs of a recession for those countries that are lucky enough to attract workers to their labour supply.

“Immigrants take our jobs”

A very right wing argument and once again simply not true. To be fairly crude, lets separate migrants into skilled and unskilled workers. You may be surprised to know that 40% of all UK migrants actually hold a university qualification, which is much better than for the rest of the EU. A large majority hold at least a vocational qualification or similar. Skilled workers are highly demanded around the world for obvious reasons and any country would be extremely jealous of the UK’s influx of such a precious commodity. Skilled workers can only add to the economy whether it is through better products, new skills, a new comparative advantage or greater competitiveness of the UK. Either way, a more productive workforce can only increase the standard of living in an economy.

In the UK, construction workers are in fact in very short supply. With rising demand for property, especially with the government’s new Help to Buy scheme, the UK needs more construction workers to fill the supply gap. Without immigration, rising demand may be met with higher prices rather than a larger supply.

As for unskilled workers, most end up in roles that pay near minimum wage. These are jobs that locals are unwilling to take. It is not the case that unskilled workers are pushing down wages and replacing UK natives. Locals are simply not willing to work at the going wage rate. If this is the case, then I see no reason why UK companies should hire equally or less skilled natives for more money than they are worth. If an unskilled worker refuses to work at the competitive wage then the UK needs unskilled immigrants to fill the gap.

“Abuse our educational system”

This one is less frequent but completely illogical. Education is one of the UK’s greatest exports and I strongly believe that it is a grave mistake to limit student visas. Foreign student fees are much higher than Home fees. It is thanks to a large inflow of foreign students that universities are allowed to flourish and stay afloat. Even if a foreign student moves on after her education, she has still invested a large amount of money in the UK educational system.

Furthermore, it is often the case that foreign students stay in the UK and go on to become highly skilled workers here. This is a win-win for the UK. We charge these students extortionately high prices and they then go on to pay taxes and add to the UK economy afterwards. In a highly globalised world, the country with the most productive workforce will usually prosper.

“Too many people”

The idea that immigration leads to overpopulation is misleading. Net immigration, after the EU was established, averaged at 214,000 per year up until 2011. It then hit 165,000 in 2012, which is a modest 0.3% of our population. This is a lower level than Italy or Spain. With an ageing population and lower birth rates than previous decades, this inflow will hardly have a substantial effect on overcrowding. Thanks to large outflows as well, the UK population will not grow any faster than the rest of the world.

As for language, I find it amusing that people become agitated that an immigrant’s first language may not be English. English is still the second most widely spoken language (behind Mandarin) and the most used language in the business world. It is not going anywhere. Immigrants moving to the UK must learn the language to prosper (and be granted entry). So what if it is their second language?

Conclusion

My conclusion in simple. Increase net migration and focus on more pertinent issues. If abuse of the welfare system or the NHS is a concern, then improve controls and the efficiency of the system. Prevent the ease of tax avoidance of the wealthy and encourage new specialisations. Continue to promote the UK as an exporter of education. These will all help to improve future public finances.

Politicians know the benefits of immigration but capping it wins votes. If only politicians could ignore the ignorance, the UK economy would be much better off in the long run.