The Bankers’ Dilemma: Explaining Self-Fulfilling Financial Crises with Game Theory

Despite the best efforts of the world’s policymakers and central bankers, financial crises seem to be a fact of life in modern, capitalistic economies. While they are often very complex in their entirety, at their core, all financial crises are caused by fairly simple interactions between economic actors. Because of this, we can explain how financial crises arise through self-fulfilling expectations using game theory – a field of study pioneered by polymath John von Neumann to solve the problem of infinite regress when examining the decisions and actions of others (Harrington, 2009, 2.)  To do so, game theory “attempts to determine mathematically and logically the actions that ‘players’ should take to secure the best outcomes for themselves in a wide array of ‘games’” (Dixit and Nalebuff) or In other words, ‘how people interact with each other in different situations to secure the outcome that is best for them.’ As we will see, the actors are not always able to secure that best outcome. This piece will look specifically at four types of crises: bank runs, banking crises, currency/exchange rate crises, and sovereign debt crises.

Bank runs occur when something triggers most of a bank’s depositors to withdraw their money all at once, depleting the bank’s liquid reserves. Banking crises are failures of an entire banking system, usually marked by widespread bank runs. The Great Depression is a perfect example; the crash of 1929 hurt balance sheets, fermented distrust in the banks, and pushed people to keep their money under a mattress instead of in a bank.

Currency crises are precipitated by doubts over a government’s willingness or ability to maintain a fixed exchange rate, and subsequent speculative attacks on the currency, which drain a government’s foreign reserves. The 1994 Tequila Crisis in Mexico is one such crisis. A slight devaluation of the Mexican peso fermented doubt about the government’s willingness to uphold the exchange rate, triggering massive capital outflows, and eventually putting enough pressure on the peso that the government was unable to uphold its value.

Finally, debt crises occur when governments look like they may be unable to pay their debts. An example would be Ireland’s recent debt troubles in which government intervention to stop a banking crisis pushed government debt up to seemingly unsustainable levels, in turn pushing up Ireland’s borrowing costs and requiring intervention from the EU and IMF.

Before we begin to analyze these crises, we must understand some basic game theory. More specifically, we must understand the Prisoners’ Dilemma – a classic game first envisioned in 1950 by Rand Corporation game theorists, Merrill Flood and Melvin Dresher, as a way to model nuclear strategy. The game was then formalized in its current form by Albert Tucker to make it applicable to more fields, including the study of financial crises (Kuhn, 1997).

In Tucker’s conception of the game two people have just robbed a bank. They manage to make a getaway and hide the money before being rounded up by the police as suspects, weapons in hand. The police know that they do not have enough evidence to convict the men for the robbery, but an especially sharp detective comes up with a plan. He will split the two up, and question each individually. He will offer each a deal: rat on their accomplice, and be allowed to walk. If both of them keep their mouths shut, then the police will not have enough evidence to convict them for the robbery, and they will only be prosecuted for illegal weapon possession – a far lesser charge. If one rats out the other, the informant will walk, while the informed-upon will get 20 years. If both of them take the deal, they each get ten years – robbery charges slightly reduced for cooperation. What is the best course of action? It seems counterintuitive, but the best course of action is to talk. Let us look at it visually.

 

Fig. 1 Prisoner Two
Prisoner One Talk Don’t Talk
Talk (-10,-10) (-20,0)
Don’t Talk (-20,0) (-1,-1)

 

While the best-case scenario for the suspects is neither of them talking, each of them will have a large temptation to talk. To find this out, let us imagine for a second that prisoner two will always talk. In that case, prisoner one is left with a choice between twenty years behind bars or five years behind bars. If he is rational, which we assume he is, then he will choose five years. Now, let us imagine that prisoner two will always stay quiet. In that case, prisoner one is given a choice between no time behind bars if he talks, and a year if he stays loyal to his companion. Once again, he is better off if he talks. This little thought experiment shows us that talking is a ‘dominant strategy – one that always results in a higher payoff than the alternative, regardless of the other player’s choice. Thus, (Talk, Talk) is the Nash Equilibrium of the game. Named after mathematician and game theorist John Nash, the Nash equilibrium is the point at which each player is responding optimally to the actions of the other players (Harrington, 2009, 90).  

This game has many applications outside of the simplified criminal justice system. It is commonly applied to economics, and more specifically, the decisions that lead to financial crises.

In a very abstract sense, a prisoners’ dilemma is a model of a coordination failure. Because they cannot communicate, the parties cannot coordinate to take the strategy that will give them the highest payoff together (neither person talks), and thus will go for the best individual payoff (talking) which makes each of them worse off than if they had both stayed quiet. Bank runs exhibit this same type of failure. In a bank run, a bank’s depositors all rush to get their deposits out at once. Because banks operate with fractional reserves, they will not have enough money to redeem all of their deposits at once. Therefore, even a solvent but illiquid bank (one where assets equal liabilities, but assets cannot be easily turned into cash) can turn into an insolvent one (in which liabilities are greater than assets) in the face of a bank run. Again, let us look at this visually.

Say Alice and Bob are depositors, each with $100 in the bank. The bank lends out a portion of its deposits, say 75%. The bank is left with a balance sheet like this:

Fig.2

Assets Liabilities
Outstanding Loans: $150 Deposits: $200
Cash: $50

 

 

This bank has a perfect balance of assets and liabilities, but can only pay depositors from its cash. The bank does not expect to have to pay out more than a combined $50 to its depositors, and in most cases, won’t have to. But say there is some bad economic news that makes it seem like the bank may not be able to meet its obligations. That may prompt Alice and Bob to take their money out of the bank. This creates a game looking something like figure 3.

 

Fig.3 Bob
Alice Withdraw Don’t Withdraw
Withdraw (Q, Q) (P, L)
Don’t Withdraw (L, P) (100,100)

 

Let us take this game apart. If neither Alice nor Bob withdraw their money, then each of them will still have $100 in the bank, thus, each of them gets a payoff of 100. If either of them withdraws money, the bank will be unable to pay them the full $100, and will have to sell some of its assets. To sell them quickly enough to make payments, the bank will most likely have to sell at below-market rates. Thus, if one depositor withdraws and the other does not, the withdrawer will get some amount of money P. It is possible that with only one person withdrawing, the bank will be able to raise enough money to pay them, so P≤100. The non-withdrawer will get the short end of the stick in this case, as they will be left with the bank’s leftovers, L, when L

This is where a bank run begins to look a lot like a prisoners’ dilemma. Both Alice and Bob would be better off if neither of them withdrew money. However, if one of them withdraws money, and the other does not, the non-withdrawer will have a lot of their wealth wiped out. Knowing this, both of them will withdraw if they think the bank will be unable to pay them. Despite this Nash equilibrium being hurtful for both depositors, in it, both Alice and Bob are responding best to each other’s actions by protecting themselves from major losses by taking lesser losses. This is how bank runs can be self-fulfilling crises – if depositors are worried that there will be a run on a bank, they will all try to get their money out as to not be the last person in line: the very definition of a bank run.

Bank runs are the smallest, simplest types of financial crises, but they are a microcosm of the other types of financial crises, all of which are triggered by the same type of self-fulfilling thinking. This is most clearly seen with banking crises. Where a bank run is a mass withdrawal of deposits from a single bank triggered by uncertainty over that bank’s solvency, banking crises are large-scale losses of confidence in an entire banking system often triggered by large falls (or expected falls) in asset value, and usually involve bank runs. In short, banking crises are scaled-up bank runs involving more people, more banks, and assets other than bank deposits. Because of this, the model for a banking crisis looks just like that for a bank run. The only notable difference is that in the case of a crisis, Alice and Bob may not be depositors with bank deposits – they could be investors with a much wider array of assets and financial instruments. With that in mind, let us move on to currency crises.  

To maintain a fixed exchange rate, a country’s central bank must buy and sell its currency as to maintain its value. If Mexico wants to peg the value of the peso such that ₱10=$1, then the Mexican central bank will buy or sell pesos on the market to get the peso to that value. If the peso starts rising in value relative to the dollar, then the Mexican central bank will respond by selling pesos – increasing the supply of pesos to decrease the price of pesos in dollars (AKA the dollar-peso exchange rate.) Because Mexico can create as many pesos as it wants, decreasing the value of the peso is easy. Propping up the value of pesos is much more difficult. To stop the peso from falling in value, the central bank must buy pesos with its foreign reserves (the central bank’s stock of foreign currency) essentially increasing demand for pesos to prop up the price. Sustained downward pressure on the peso will drain the central bank’s foreign reserves. Because the Mexican central bank cannot create more foreign assets, resisting sustained downward pressure on the peso will drain Mexico’s reserves until they run out, at which point Mexico must either abandon the fixed exchange rate or impose capital controls.

Currency crises like the one described above are also often the result of self-fulfilling expectations. In fact, it was currency crises that were first noted to be self-fulfilling by economists such as Maurice Obstfeld and Paul Krugman (Krugman, 2015.) For our (much simpler) analysis, let us continue to use the example of Mexico. This time, let Alice and Bob be American investors with large positions in a certain peso-denominated Mexican asset. If the asset is worth ₱100 per unit, then Alice or Bob could sell one and be 100 pesos richer. However, ₱100 is not worth much to anyone outside of Mexico, so Alice or Bob would presumably want to exchange those pesos for dollars. At the above-mentioned 10:1 exchange rate, the asset would be worth $10. However, if the peso were to devalue, then the asset would be worth less in dollar terms. Say the peso’s “true” value (its value were the Mexican central bank to let it float) was 20 pesos per dollar. That would mean that if the peso were to float, the assets that Alice and Bob bought for $10 would be worth the same in pesos (assuming no change in asset price) but would only be worth five dollars. Because of this risk, investors may choose to pull their investments out of a country if they believe the currency will devalue. In our example, as investors sell their peso-denominated assets, they will receive pesos, which they will exchange for dollars. In other words, they start a run on a currency. Once again, let us construct a game to model this situation.

 

Fig. 4 Bob
Alice Pull Keep
Pull (Q, Q) (10, L)
Keep (L, 10) (10,10)

 

If Alice and Bob are worried that the peso will devalue, they are faced with a choice to pull their money out of Mexico, or to keep it in Mexico. If both Alice and Bob keep their money in Mexico, then no devaluation will happen, and their assets will still be worth $10 a unit. If one investor keeps their money in, and the other pulls it, then the investor who pulled their money will get sell their assets and exchange their pesos for dollars at 10:1. Depending on how well Mexico’s reserves hold up, they may or may not be able to maintain the exchange rate as the first investor pulls out. Thus, the investor who initially kept their money in Mexico will get a payoff of L, where L≤10. If Mexico’s reserves hold out, then L=10, but if they do not, then the peso will devalue, and whoever kept their money in Mexico longer will end up on the bad end of the run; their pesos will devalue as they sell them. If both investors withdraw, then the peso will devalue more quickly, but both investors will be able to get some of their assets out at the 10:1 exchange rate, so each of them will get a payoff of Q, where A

This assumes that Mexico was not planning on devaluing; however, the game would not be significantly different if Mexico was going to devalue no matter the actions of investors.  The major differences are that the payoffs for (Keep, Keep) would be lower than 10, as would the payoff for Pull in the cases of (Pull, Keep) or (Keep, Pull.)

Once again, we are left with something that looks very much like a prisoners’ dilemma. Both investors would be the most well off if they both stayed in Mexico. However, each one worries that the other will sell. So, with neither wanting to be late to the sell-off, both investors will end up selling.

It is worth noting that in this example, Alice and Bob could just as well be Mexican citizens with little faith in their currency. If they were to begin worrying that the peso will devalue (making them, in effect, poorer) they may choose to buy dollars, which will put pressure on the peso to devalue. In fact, if foreign investors were to see evidence of such dollarization, it may prompt them to begin selling off pesos. Conversely, speculative attacks by foreigners might prompt dollarization. Either way, the reactions of ‘players’ in this situation can turn the threat of a crisis into an actual crisis.

The final type of crisis we will look at is the sovereign debt crisis, but before we delve into how a government can become unable to pay its debts, let us look for a second at how government debt works. Governments take in money by levying taxes, which we will denote with “T.” Governments then spend this money on things like the military, police, infrastructure, pensions, subsidies, and healthcare. We will denote this government spending with “S.” In this model, a government must spend exactly what it takes in such that T = S. However, one need look no farther than the front page news to know that governments don’t always adhere to this rule. When governments spend more than they take in, they must resort to borrowing, which we will denote with B. Thus, a more realistic government budget constraint looks like this:

S=T+B

However, that budget constraint only works for the first period “t.” Once the government has borrowed, it will have to start paying interest – “r” – on that borrowed money. We denote this debt as D, where D is equal to the sum of all unpaid debts from previous periods. Thus, the budget constraint over multiple time periods will look like this:

St+rD=Tt+Bt

So say that the year is 1829, and Greece has just won its independence from the Ottoman Empire. We will assume that this new country starts with no debt. In its first year of independence, the Greek government spends 100,000 drachmas, but brings in only 90,000 in taxes. It must borrow 10,000 drachmas to pay for its operations, so it does so at a 10% interest rate. In this example, S1=100,000; T1=90,000; and B1=10,000. Because there was no Greek government to borrow before 1829, there is no old debt to pay off, so Bt-1=0. In period two, say the government takes in the same amount in taxes, and spends the same amount. However, in period two, the government must also pay off interest on its period one borrowing, which will be 10% of 10,000, or 1,000 drachmas. Because the Greek government must pay this debt, and has not gotten any more in tax revenue, it will have to borrow further to pay the interest. The second period budget constraint will look like this:

100,000+1,000=90,000+11,000.

As the Greek government continues to run a deficit (T<S) its stock of debt will increase, as will its interest payments in the next period. In the model so far, Greece has no other option but to keep borrowing more to pay interest on previously-accrued debt, but a quick survey of history will show that not to be the case. Countries can, and do default on their debt. In terms of our simple model, a default would be Greece refusing to pay interest on accrued debt. Even though defaulting is usually the end result of a complex cost-benefit analysis on the part of the government, the amount of debt that the government carries is a major determinant as to whether or not a country will default (Reinhart and Rogoff, 2009.) Even more importantly for us, the amount of debt a country carries (usually expressed as a ratio of debt to GDP) is a highly visible indicator that creditors can point to. If these creditors see Greece piling on more and more debt, they will see Greece as more and more of a risk. As compensation for the additional risk, Greece’s creditors will begin to ask for higher interest rates on Greek debt. This creates a feedback loop in which more debt leads to higher interest rates which require more borrowing to pay for, which leads to more debt. It also introduces the possibility of a fragile equilibrium – a concept we have seen with banking and exchange rate crises, but have not formally defined.

In its most basic sense, a fragile equilibrium is one that will hold absent any shocks, but that will shift in the face of a shock. A good example would be a ball balanced on top of an upturned bowl. The ball will balance on the domed surface in calm conditions (the fragile equilibrium) but a gust of wind (the shock) could easily blow the ball off the bowl and onto the surface below (the new, less-fragile equilibrium.) This is not to say that the post-shock equilibrium is necessarily permanent (although it very well could be.) It will just take more effort to move the ball back to the top of the bowl. In the example of the bank run, we have depositors who will keep their money in the bank (or withdraw it in small amounts) in normal times, but if a shock comes along, they will all withdraw at once. Everyone keeping their money in the bank is the fragile equilibrium. People will keep their money in the bank absent a shock, but shocked people will try to withdraw every dollar they have. A similar situation can arise with sovereign debt.

Let us revisit Greece to see how this could work. Say that Greece has some amount of debt that is high, but considered sustainable by its creditors. It pays a 5% interest rate on this debt, and has no trouble paying at this rate. However, if the interest rate were to go up significantly, then Greece would have difficulty paying its debt, and would be much more likely to default. The game for this crisis is significantly more complicated than were the games for banking and currency crises.The Bankers Dilemma - Cooper.docx

In this game, Alice and Bob are Greece’s creditors, who can either buy or not buy Greek bonds. If both creditors choose to buy bonds, then each of them gets a payoff of Ar, which is the perceived benefit of buying bonds at rate r. If one creditor buys and the other does not, the non-buyer gets nothing, while the buyer gets a payoff of Br, where Br>Ar, as we assume that the buyer will pick up the slack (if the buyer does not, then Greece will default, and the buyer will lose money.) In either of those cases, Greece gets a payoff of Xr, which represents Greece’s  net benefits from making continued payments. In other words, the benefits of continued borrowing less the costs of paying interest. If neither buys, Greece gets an opportunity to adjust the rates on its bonds. It can either offer bonds for lower prices (raising their yield,) or default.

If Greece chooses to lower its bond prices, then the game repeats with different values for Ar, Br, and Xr. If the new prices are enough compensation for the extra perceived risk of holding Greek debt, then the creditors will buy bonds again, and the game will end for this period. If the first offered rate is not enough of a risk premium, then the game will repeat again. If Greece defaults, then the creditors will get payoffs of D, which represents the losses they incur from holding previously-bought bonds that are now worthless (it makes little sense to only default on some debt, because the consequences of default will be the same as on a total default.) Yr represents Greece’s net benefit for defaulting at the current rate – essentially the benefit of shedding debt and interest payments minus the costs of becoming shut out of international capital markets and possibly having foreign assets seized. At the game’s outset, we will assume that making continued payments is a better deal than defaulting. Mathematically, Xr>Yr at the beginning.

In the case that Greece was able to make its debt payments until some exogenous shock forced them to raise interest rates, it is possible that Greece could continue to make payments at the higher rate. However, this would result in a higher level of debt, which opens the possibility of Greece entering a vicious cycle. To illustrate, let us imagine that Greece is paying a 5% interest rate on its debt in the first period. In the second period, Greece’s creditors begin to think that Greece may not be as safe as they had thought it, and refuse to buy Greek bonds. In response, Greece offers 7%, which the creditors take. The resultant increase in debt then makes Greece’s debt look even less sustainable, and therefore riskier, so creditors demand an additional risk premium, raising Greece’s debt again and making the debt still less sustainable and still riskier. All this involves the assumption that Greece’s creditors do not know the exact values of Xr and Yr, but do know how they behave. As the interest rate climbs, default will begin to seem like a better option for Greece than continuing to make ever more tortuous payments. At some point, Greece will actually gain more (or lose less, as the case may be) by defaulting than by paying. In the model’s terms, each time Greece lowers prices on its bonds, Xr goes down and Yr goes up. When Yr>Xr, Greece will default.

The idea that a country can gain more by defaulting than by paying its debts seems far-fetched at first. However, when one looks at the hardships countries go through to pay their debts, it begins to look far more realistic. Paying down debt involves raising taxes and cutting spending (which includes benefits to citizens) – neither of which are winning propositions for the population. 1980s Romania provides a perfect example of a country that refused to default, and instead paid its debts at all costs. Most Romanians ended up with no heat during the winter, and the country was chronically short of electricity (Reinhart and Rogoff, 2009, 51.) While hardship like this may have been enforceable in communist Romania, they would not be in most countries. A government with less control over its people that forced its people to endure harsh winters with no heat or electricity would quickly find itself forced out. Thus, default can become the more appealing option if a country finds itself trapped in the vicious cycle of a debt crisis.

We have now seen how self-fulfilling expectations can cause a crisis. Rumors about a bank’s or banking system’s insolvency can prompt mass withdrawals that make banks insolvent. Worries that a country may devalue its currency can invite speculative attacks and force a devaluation. And the risk of sovereign default can raise a country’s interest payments until that default becomes a reality. We will now briefly examine how these types of crises can interact.

The possibility of a crisis is always there, but the nations of the world are not always in a state of crisis. This is because these crises are, more often than not, triggered by some sort of a shock. These shocks can come in many forms: the end of an asset bubble, a natural disaster, and war, among others. However, the different types of crisis can also trigger each other. To illustrate this, let us turn to the real-world example of the 1994-95 Mexican ‘Tequila Crisis.’ In a nutshell, the crisis began as a currency crisis, and turned into a debt crisis as well. In the wake of political instability in 1994, investors began to worry that Mexico might devalue the peso, and began to withdraw their investments in Mexico, putting downward pressure on the peso. To calm investors and replenish their reserves, Mexico issued debt instruments called tesobonos, which were insured against devaluation (van der Molen, 2013.) This worked for a time, but after elections in December of 1994, money began to flow out again, and after a promise not to devalue the peso fails to calm investors, Mexico devalued the peso by 15% (van der Molen, 2013.) This shattered faith in the peso, and investors withdrew more and more money, eventually forcing the peso to float, which devalued it further. This devaluation made tesobonos and Mexico’s foreign-denominated debt harder to service, as they suddenly took far more pesos to pay off than they had before. A change in the exchange rate had turned Mexico’s debt unsustainable overnight, and the IMF and US Government had to step in to keep Mexico from defaulting (van der Molen, 2013.)

This is just one example of how crises can cause each other, but it is not difficult to see other ways that it could happen. For example, a devaluation might make banks’ foreign-denominated debt unpayable and lead to a banking crisis as depositors try to get their money out of banks with more liabilities than assets. Conversely, a banking crisis caused by something like an asset bubble popping could prompt foreign investors to sell their assets in a country, putting pressure on the domestic currency until the central bank can no longer maintain the exchange rate.

However, there is one last thing to keep in mind about financial crises, illustrated quite well by the American-IMF rescue of Mexico. The game theoretical models I have set up to examine the mechanics of financial crises make these crises seem inevitable, when they may not necessarily be. Each type of crisis is the product of self-fulfilling expectations. Without those expectations being fermented, there is no reason for economic actors to push into a crisis. In the case of a bank run, there is little reason in most circumstances to withdraw all of one’s savings from a bank, but the moment that everyone begins worrying that their money is not there, and starts to make a run on the bank, it suddenly becomes reasonable to do the same. The threat of a bank run is always there, but the slightest observation will tell you that there are not bank runs going on constantly, so it seems that people can control their expectations absent a shock. This is helped by policies and institutions in place to prevent such crises, although they still occur. Most countries have deposit insurance to combat banking crises. The IMF was originally conceived as an institution to ferry reserves to countries with imperiled exchange rates, and has recently expanded into rescuing indebted countries. While these may bolster confidence and prevent expectations from pushing into a crisis in some cases, they have not ended economic crises. They may not be inevitable, but they have not been eradicated either. And they may not be, so long as – in the words of John Maynard Keynes – “The market can stay irrational longer than you can stay solvent.”

 

Notes:

Special thanks to Bryan Pratt for helping me with the debt crisis model.

 

Works Cited:

Dixit, Avinash, and Barry Nalebuff. “Game Theory.” The Concise Encyclopedia of Economics. Library of Economics and Liberty. Web. 30 July 2015.

Harrington, Joseph Emmett. Games, Strategies, and Decision Making. New York: Worth, 2009. Print.

Krugman, Paul. “A Self-Fulfilling Euro Crisis? (Wonkish).” New York TImes 7 Aug. 2015, Opinion sec. Web. 17 Aug. 2015. <http://krugman.blogs.nytimes.com/2011/08/07/a-self-fulfilling-euro-crisis-wonkish/>.

Kuhn, Steven. “Prisoner’s Dilemma.” Stanford Encyclopedia of Philosophy. Stanford University, 4 Sept. 1997. Web. 17 Aug. 2015.

Reinhart, C., & Rogoff, K. (2009). This time is different: Eight centuries of financial folly. Princeton, New Jersey: Princeton University Press.

Van Der Molen, Maarten. “The Tequila Crisis in 1994.” Rabobank. Rabobank, 19 Sept. 2013. Web. 17 Aug. 2015.