Every now and again someone will ask what I’m working on. Occasionally, these people actually seem to be genuinely interested (rather than just being polite!). This post is for those people. Ryan Oprea and I have recently submitted our joint paper to a journal, which means that it is now available on the web and ready for public consumption.

The paper, Continuity, Inertia and epsilon-Equilibrium: A Test of the Theory of Continuous Time Games, is an investigation of how people make strategic decisions under various timing protocols.

There are two main ways to conceive of time: either as discrete time (where time advances in discrete chunks) or continuous time (where time advances continuously). Generally, we think that continuous time is a more realistic way of viewing the world.

However, mathematical models of behaviour in continuous time have some strange properties. Because time unfolds continuously, agents can react instantly to changes in their environment. Obviously, this is not the case for real life behaviour. This distinction between the model of continuous time (which we call Perfectly Continuous time) and the reality of continuous time (which we call Intertial Continuous time) has meant that previous economic experiments that have tried to test continuous time models have generally produced mixed results: the reaction lags inherent in Inertial Continuous time mean that they were testing the wrong model. 

The first thing that we set out to do in our paper was to perform a genuine test of a Perfectly Continuous time model. But to do this, and bring reality into line with the model, we need a way to eliminate human reaction lags [1]. We used a very simple trick: we had subjects play an exceptionally basic “computer game” with each other and whenever something in the environment changed we instantly (and automatically) froze the game clock. While the game clock was frozen the subjects could decide about how they wanted to respond (and enter their choice into the computer). Then, when the clock was unfrozen, the subject’s response was enacted immediately. In this fashion, subjects can make decisions instantaneously in game time.[2] In all, we had three different timing setups. Perfectly Continuous time (with the freeze protocol), Inertial Continuous time (without the freeze protocol) and Discrete time (time is cut up into chunks).

The game we gave subjects was very simple, but was designed to generate stark predictions under the different timing protocols. Subjects were matched into pairs, and each subject in a pair had to decide when to click a button and “enter the market”; the subjects were paid depending on when the subjects entered. The subjects earned more the longer they waited before entering, but were also rewarded for being the first person to enter (and penalized if they were the second person to enter). There is, therefore, a tension between wanting to delay to earn more money, but not delaying for so long that your opponent enters before you.

The standard economic model of Nash equilibrium (originally proposed by John Nash) predicts that, in Discrete Time and Inertial Continuous time, everyone should enter immediately at t=0. However, in Perfectly Continuous time there are many Nash equilibrium, and the one that produces the best outcomes for the subjects predicts entry after 40% of the period has elapsed.

Based on previous experimental results (some of which Ryan Oprea had conducted himself) we were pretty confident that the Nash equilibrium model would perform poorly for the Inertial Continuous time case and possibly also for the Discrete time case. We hypothesized that an alternative model, epsilon-equilibrium, which allows for subjects to make “small” mistakes would perform better. The problem with epsilon-equilibrium is that it does not make very precise predictions. It does, however, predict that when reaction lags are very big then everyone should enter at t=0 (the same as the Nash case), but when reaction lags are smaller then some level of delay before entry can be supported. Additionally, it also predicts that, in Discrete time, when the reward for entering first is very big then everyone should enter at t=0 (the same as the Nash case), but when the reward is smaller then some level of delay before entry can be supported.

This lead to the problem of how to manipulate reaction lags: because reaction lags are naturally occurring in our subjects, we cannot influence them directly. However, the theory tells us that what is important for our game is the reaction lag relative to the speed of the game. So we ran a Fast treatment where we sped up our Inertial Continuous time game, and a Slow treatment where we slowed it down. Our initial Inertial Continuous time treatment lasted 60 seconds, while our Fast was 10 seconds (really fast!) and our slow was 280 seconds.  We measured reaction lags at about 0.5 seconds in all three treatments, so that our relative reaction lags varied from about 5% of the game length in the Fast treatment to about 0.2% of the game length in the Slow treatment.

Now, onto the results!

In the Perfectly Continuous time treatment, pretty much everyone entered at (or very, very close to) 40% of the way through the period, exactly like Nash equilibrium predicted. This is a strong vindication of the theory of continuous time games – when reaction lags are eliminated from behaviour, then the theory does very well!

In the Inertial Continuous time treatments, the Nash model did pretty poorly. In the Fast treatment, the median entry time was t=0, but in the baseline treatment median entry was 20% of the way through the period and in the Slow treatment median entry was 30% of the way through the period. These results are consistent with epsilon-equilibrium, but not Nash. The results from the two Discrete time treatments were also consistent with epsilon-equilibrium but not Nash.

There was another interesting facet to the data. The theory of epsilon-equilibrium suggests, with a wink and a nudge, but does not guarantee, that as reaction lags become very large that Inertial Continuous time behaviour should approach Discrete time behaviour and that as reaction lags become very small then Inertial Continuous time behaviour should approach Perfectly Continuous time behaviour.

This pattern is neatly traced out in our data, and this has implications for the way that we, as economists, should model behaviour. For interactions with very rapid response (such as online high frequency trading, where lags are a fraction of a second), models in Perfectly Continuous time should provide a good approximation of behaviour. On the other hand, for interactions with longer responses (such as the back and forth negotiating over a house purchase, where responses can take up to a few days) then Discrete time models should provide a good approximation of behaviour. The more realistic, but much more difficult to work with, model of Inertial Continuous time is not really necessary to produce good predictions.

So, there you have it. After a bit over 18 months of working on this project, we have a working paper that has been submitted to a journal. And given the way that the publication process works in economics, it could be anywhere from 6 months to 7 plus years before this paper gets published anywhere!

[1] Now, you might well be wondering why we would want to test a model that is clearly not an accurate description of how reality works. The answer is that Perfectly Continuous time models seem to work well in some settings, but not in others. The more we understand about why the model performs poorly in some domains, then the easier it is to build better models.

[2] In a sense, this is reminiscent of the old James Bond game on playstation  where you could switch into “Bond Mode” where the whole world would slow down so that you could think through your strategy without getting shot at the whole time. We did basically the same thing, only we completely froze the game instead of just slowing it down. Plus instead of having an awesome first person shooter we just had two dots moving along three different lines and the only action you could take was to click a single button. But otherwise it was totally like a James Bond game.