Measured Games · Paper 04 of 07 · Econophysics

The minority game

After W. B. Arthur (1994) and D. Challet and Y.-C. Zhang (1997); phase structure from R. Savit, R. Manuca and R. Riolo (1999)

Abstract

Agents rewarded for joining the smaller of two groups cannot hold a shared correct expectation, since any widely held forecast invalidates itself. Replacing deduction with inductive strategy selection produces a system whose collective efficiency depends on a single parameter, the ratio of strategy space to population, and which shows a sharp transition between a herding regime worse than random and an efficient regime better than it.

Keywordsinductive reasoning · anti-coordination · phase transition · market efficiency

1Setup

Setup

Arthur (1994) posed it as the El Farol bar problem. A hundred people each decide independently whether to go to a bar on Thursday night. The bar is enjoyable if fewer than sixty turn up and unpleasant if more do. Nobody communicates. Everyone can see the attendance history.

The trap is that no shared rational model can survive. If everyone believed a model predicting light attendance, everyone would go and the model would be wrong. If everyone believed the opposite, nobody would go and it would be wrong again. Any deductive reasoning process, applied by everybody, destroys its own premise. Arthur’s point was that this is a general condition wherever beliefs determine outcomes: deduction has nothing stable to work from, so agents must reason inductively, holding several hypotheses and using whichever has been working.

2The model

The model

# minority game, Challet & Zhang (1997) N agents (odd) choose -1 or +1 each round A(t) = Σ ai(t) the minority side wins each agent holds S strategies: a lookup from the last m outcomes → a choice all strategies scored on what they would have done the agent plays its current best scorer # the single control parameter α = 2m / N strategy space per agent σ2/N volatility; = 1 for coin flipping αc ≈ 0.3374 (for S = 2) α < αc symmetric phase: no exploitable signal, high volatility α > αc asymmetric phase: signal present, volatility nearer random

Below the critical point the agents draw on too few distinct strategies, herd on the same signals and swing wildly. Above it their behaviour decorrelates. At the transition the crowd is at its most efficient and there is no exploitable structure left in the history.

Figure 1Inductive agents and the volatility transition

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Attendance A(t) · deviation from an even split

Distribution of attendance · measured against a coin-flipping population

Volatility against α · press Sweep α to trace the curve. Hover to read values

Attendance Coin flipping benchmark Measured volatility Current setting Observed spread Coin-flipping spread

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Agents N

Memory m

Strategies per agent S

α = 2^m/N—

Volatility σ²/N—

vs coin flipping—

Phase—

Winners this round—

Round0

Figure 1. Every agent holds real strategy tables and scores them on counterfactual performance, so the volatility readout is measured rather than looked up. Sweep α runs eleven memory lengths for 4,000 rounds each and plots the result: the curve dips below the coin-flipping benchmark and comes back, with the minimum near α = 0.34. Left of the dip the crowd is doing worse than random.The middle panel compares the observed spread of attendance against what a coin-flipping population of the same size would produce. In the crowded phase the observed distribution is visibly wider, which is the herding the volatility number is measuring.

3Results

Results

Savit, Manuca and Riolo (1999) found that the behaviour of the whole system collapses onto the single parameter α, the size of the strategy space relative to the number of agents. Volatility, measured as the variance of the difference between the two sides, has a minimum at a critical value near 0.34, with two genuinely different phases either side of it.

Below the critical point the agents draw on too few distinct strategies. They herd, act on the same signals, and the crowd swings harder than if everyone had simply flipped a coin. Volatility falls as α rises, reaches a minimum well below the coin-flipping level at α_c, then climbs again and approaches the coin-flipping limit from below as α grows large.

The two phases either side of the minimum are usually named for what they do to information rather than to volatility, and the two run in opposite directions, which is the part that is easy to get backwards. Below α_c the system is in the symmetric or unpredictable phase: both choices win with equal probability after any given history, so nothing in the record can be exploited, even though the crowd is performing badly. Above α_c the system is asymmetric and predictable: the history does carry a signal, and an agent able to read it could profit, even though aggregate performance is closer to random.

That inversion is what makes this a market model rather than a puzzle. Informational efficiency, in the sense of no exploitable signal, is bought with high volatility rather than with calm, and the model produces that trade-off from boundedly rational agents rather than assuming it.

4Evidence

Evidence

The empirical grounding here is different in kind from the biological models on the other sheets. Nobody found a minority game in a field site. The claim is that a minimal set of assumptions reproduces the statistical signatures of real markets, and it does.

Challet, Marsili and Zhang (2005) develop the mapping to price formation, in which the majority side moves the price against itself and the minority profits. Grand-canonical versions, where agents can decline to trade, generate fat-tailed return distributions and volatility clustering, the two most robust stylised facts in market data, with no exogenous shocks. The critical point corresponds to the boundary between a predictable market and an efficient one, and the model predicts that markets sit near it because agents enter while predictability lasts and stop when it does not.

Johnson, Jefferies and Hui (2003) collect the wider body of work applying this class of model to financial time series.

5Limitations

Limitations

Reproducing stylised facts is weaker evidence than a controlled measurement, and it should be said plainly: several unrelated models generate fat tails and volatility clustering. What the minority game adds is a mechanism that also explains why a market would sit near the critical point rather than anywhere else, which is a stronger claim than matching a histogram.

The agents are also very simple. They hold fixed strategy tables, never invent new ones, never learn from other agents and have no wealth constraint, none of which is true of traders. Extensions that add these features change the location of the transition and sometimes remove it.

Read this model as a demonstration that anti-coordination alone can produce market-like statistics, not as a description of how any particular market works.

References

Arthur, W. B. (1994). Inductive reasoning and bounded rationality. The American Economic Review, 84(2), 406–411.
Challet, D., Marsili, M., & Zhang, Y.-C. (2005). Minority games: Interacting agents in financial markets. Oxford University Press.
Challet, D., & Zhang, Y.-C. (1997). Emergence of cooperation and organization in an evolutionary game. Physica A: Statistical Mechanics and Its Applications, 246(3–4), 407–418. https://doi.org/10.1016/S0378-4371(97)00419-6
Johnson, N. F., Jefferies, P., & Hui, P. M. (2003). Financial market complexity. Oxford University Press.
Savit, R., Manuca, R., & Riolo, R. (1999). Adaptive competition, market efficiency, and phase transitions. Physical Review Letters, 82(10), 2203–2206. https://doi.org/10.1103/PhysRevLett.82.2203

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