Effects of Tobin Taxes in Minority Game markets

Ginestra Bianconi; Matteo Marsili; Tobias Galla

REVIEW

Effects of Tobin Taxes in Minority Game markets

Ginestra Bianconi, Tobias Galla, Matteo Marsili

Disordered Systems and Neural Networks · Submitted 6 Mar 2006 · 25 references

Pith's verdict: A solid, analytically grounded MG study of Tobin taxes; the volatility-reduction result holds up, but the revenue-maximization claim in the conclusions is overstated for total revenue.

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CitationGinestra Bianconi, Matteo Marsili, Tobias Galla. "Effects of Tobin Taxes in Minority Game markets." arXiv:cond-mat/0603134 (2006). https://pith.science/paper/LEILHWUW

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The paper to know: Bianconi, Galla, and Marsili apply the grand-canonical Minority Game to Tobin taxes. The genuinely new result is that a transaction tax reduces volatility only near the GCMG's critical line — i.e., in markets with anomalous fluctuations — and has little effect far from criticality. This comes from the exact replica solution (Eqs. 11–13), not from fitting, and the simulations at L=3000–12000 with at least 1000 samples line up with theory outside the critical region and deviate systematically inside it, exactly as expected for finite-size effects. That part is solid.

Now the soft spot. The stress-test is correct. Eq. (7) defines total revenue R = Rs + Rp, where Rp = τ np/(ns+np) is linear in τ and unbounded as τ grows because producers always trade. The speculator component Rs peaks, but total R does not. So the conclusion's claim that "the revenue for the market maker from the tax attains a maximal value at intermediate tax rates" is false for the total revenue as defined. The abstract correctly says "revenues obtained from speculators" — that is the right qualification. Dropping it in the conclusions is a genuine internal inconsistency, and it inflates the "optimal intermediate tax rate" policy narrative. It should be fixed, but it is not fatal; the volatility-reduction result does not depend on it.

The modeling limitation the reader flagged is real: setting producers at ε=-∞ means the tax only hits speculators, and the connection to real tax rates is indeed hard. The authors state this at the end, so it is a caveat, not a hidden flaw. It lowers significance more than soundness.

Is this a serious thinker's paper? Yes. The analytical machinery is used carefully, the simulations are properly executed, and the text is honest about its own limits. The revenue overstatement looks like an oversight rather than a deliberate misrepresentation.

My recommendation: send it to peer review. A competent referee will catch the revenue wording and ask for the total-revenue curves to be shown with the linear producer term separated out. The paper will be better for that. The core message — tax reduces volatility near criticality and can help coordination when the population changes — is worth publishing.

Referee report

Significance

The main volatility result is well supported: the analytical formulas are not fitted, the tax enters only through the substitution ε→ε+τ, and the finite-size simulations deviate from theory in exactly the critical region where the proposed mechanism operates. The paper also gives a concrete dynamical interpretation of why a tax helps when agent turnover is slow, which goes beyond a purely static phase-diagram argument. The revenue conclusions as written are not all correct: the 'maximal revenue at intermediate tax rates' statement in the conclusions is only valid for the speculator component, and the claim about the relative sizes of speculator and producer revenues is internally inconsistent. These issues are local and correctable, and they do not invalidate the central volatility-reduction mechanism.

Major objections (2)

[Section V, Eq. (7)] The conclusion that "the revenue for the market maker from the tax attains a maximal value at intermediate tax rates" is false for the total revenue R defined in Eq. (7). Since producers always trade, the producer contribution is R_p = τ n_p/(n_s+n_p), which is strictly linear and unbounded in τ, whereas the speculator contribution R_s = τ(n_s/(n_s+n_p))φ(ε+τ) decays to zero for large τ. The total R therefore cannot possess a finite interior maximum; at most R_s does. The abstract correctly restricts the maximum to "revenues ... obtained from speculators," but Section V drops this qualification. This is not merely cosmetic, because the "optimal intermediate tax rate" narrative in the conclusions depends on the existence of such a peak; the statement should be corrected or R redefined.
[Section V (Conclusions) vs. Section III, Fig. 4] The statement in Section V that "in both shown examples the revenue for the market maker appears to weigh more on institutional investors than on speculators" contradicts the discussion of Fig. 4 in Section III, which reports for ns=60, np=1 that "the contribution to the tax revenue of speculators largely outweighs that of producers." The subsequent claim that "we find R_p > R_s (not shown here)" is also presented without supporting data in the relevant parameter regimes. Since the revenue composition is part of the paper's stated findings, this inconsistency needs to be resolved, either by correcting the conclusion or by clearly distinguishing the parameter regimes.

Minor objections (4)

[Appendix, Eq. (11)] There is a missing closing parenthesis in the expression for Q(ζ,K), which contains "erf[(1+K)ζ/√2 − erf(Kζ/√2)" instead of "erf[(1+K)ζ/√2] − erf(Kζ/√2)".
[Introduction and Section II B] The text contains small typos: "at at the same time" in the Introduction, and "It is found simulations that" should be "It is found in simulations that".
[PACS line] The line "PACS numbers: PACS" appears to be a placeholder and should be completed or removed.
[Caption of Fig. 3] The statement that the tax rate "increases from top to bottom" is easy to misread as printed; the top panel has ε+τ=0.2 and the bottom panel ε+τ=1.0, so the sentence could be rephrased for clarity.

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The paper

The paper claims

In the grand-canonical Minority Game, a per-transaction tax reduces market volatility when the market is near its critical line, with the largest effect where anomalous fluctuations already occur.

The reading

The paper shows that, in a tractable agent-based model of a currency market, a small per-transaction tax can reduce exchange-rate volatility by pulling the market away from a critical phase transition at which it is perfectly efficient. The effect is strongest close to that transition, exactly where the model produces fat-tailed returns and volatility clustering, and much weaker when the market operates far from criticality. Tax revenue is largest at intermediate rates, near the point where volatility is lowest. A second result covers markets whose trader population turns over in time: the tax helps agents coordinate faster, lowering volatility even at large system sizes.

What carries the argument

The central object is the phase diagram of the grand-canonical Minority Game, with speculator density $n_s$ and margin $\epsilon$; a critical line at $\epsilon=0$, $n_s\ge n_s^\star(n_p)$ separates an inefficient finite-predictability phase from a perfectly efficient one. The mechanism is the substitution of Eq. (6), $\epsilon\to\epsilon+\tau$ for speculators only, which makes the tax act as a uniform additive cost shift and therefore as a displacement in parameter space. The analytic backbone is the replica solution of the model, which reduces the stationary state to two coupled equations for variables $K$ and $\zeta$, from which predictability, mean activity, and an accurate approximation for volatility follow.

Extended reading notes

Core claim

Within the grand-canonical Minority Game, levying a Tobin tax is represented by replacing each speculator's trading threshold $\epsilon$ by $\epsilon+\tau$ while leaving producers' cost at $-\infty$. Because the model has a critical line at $\epsilon=0$, $n_s>n_s^\star$, the tax moves the effective control parameter away from criticality, suppressing the anomalous fluctuations that appear there. The paper shows, with simulations and with the exact statistical-mechanics solution of the model, that volatility then drops substantially when the market is in this near-critical regime, that predictability scales as $H\sim \epsilon^\alpha$ with $\alpha=2$, and that tax revenue from speculators has a maximum at intermediate tax rates. Far from criticality the tax barely moves either volatility or efficiency.

Load-bearing premise

All results follow from treating the tax as a pure additive cost shift $\epsilon\to\epsilon+\tau$ for speculators only, with producers' participation, the news process, and market liquidity unchanged; a real tax that also changes those would not be captured.

Editorial extensions

If this is right

In a near-critical market, a small tax should visibly lower volatility while the market remains close to full efficiency.
Tax revenue peaks at intermediate rates, approximately where the volatility reduction is largest, so revenue and stabilization motives align in this regime.
Far from criticality the tax has little effect, meaning a uniform tax would leave already stable markets essentially untouched.
In markets with a slowly evolving trader population, the tax shortens the coordination time scale, which scales as $1/(\epsilon+\tau)$, and can prevent sustained high-volatility states.
The volatility reduction is stronger in smaller markets, and the effect weakens as system size grows toward the infinite-size theoretical limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the mechanism reaches real markets, transaction taxes should show the strongest volatility response exactly where markets display fat tails and volatility clustering, and almost no response in calm efficient markets; that asymmetry is a testable empirical signature.
The $1/(\epsilon+\tau)$ equilibration time suggests a small tax may fail when trader turnover is faster than coordination; a natural policy extension is a tax rate scaled to turnover.
The model's tax applies only to speculators; an obvious extension is to couple the tax to liquidity $\lambda$, since a tax that drives speculators out could thin the market and offset part of the volatility gain.
The same parameter-shift mechanism might be checked in Minority Game variants with endogenous information or multiple strategies, where the critical line may move but the displacement argument should still hold.

Original abstract

We show that the introduction of Tobin taxes in agent-based models of currency markets can lead to a reduction of speculative trading and reduce the magnitude of exchange rate fluctuations at intermediate tax rates. In this regime revenues for the market maker obtained from speculators are maximal. We here focus on Minority Game models of markets, which are accessible by exact techniques from statistical mechanics. Results are supported by computer simulations. Our findings suggest that at finite systems sizes the effect is most pronounced in a critical region around the phase transition of the infinite system, but much weaker if the market is operating far from criticality and does not exhibit anomalous fluctuations.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We show that the introduction of Tobin taxes in agent-based models of currency markets can lead to a reduction of speculative trading and reduce the magnitude of exchange rate fluctuations at intermediate tax rates. In this regime revenues for the market maker obtained from speculators are maximal. We here focus on Minority Game models of markets, which are accessible by exact techniques from statistical mechanics. Results are supported by computer simulations. Our findings suggest that at finite systems sizes the effect is most pronounced in a critical region around the phase transition of the infinite system, but much weaker if the market is operating far from criticality and does not exhibit anomalous fluctuations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that the introduction of Tobin taxes in agent-based models of currency markets can lead to a reduction of speculative trading and reduce the magnitude of exchange rate fluctuations at intermediate tax rates.	result 0.99	0.93	0.89	asserted 0.82
s1	In this regime revenues for the market maker obtained from speculators are maximal.	result 0.88	0.78	0.07	asserted 1.00
s2	We here focus on Minority Game models of markets, which are accessible by exact techniques from statistical mechanics.	method 0.92	0.33	0.02	asserted 0.99
s3	Results are supported by computer simulations.	method 0.99	0.81	0.02	asserted 0.98
s4	Our findings suggest that at finite systems sizes the effect is most pronounced in a critical region around the phase transition of the infinite system, but much weaker if the market is operating far from criticality and does not exhibit anomalous fluctuations.	result 0.80	0.92	0.24	hedged possibility 0.98

What kind of paper the classifier reports this is: simulation study 0.71 · evidence from simulation or computation 0.97.

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arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

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R cross-paper relations

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Figures

**Figure 1.** FIG. 1: Phase diagram of the GCMG in the (

**Figure 2.** FIG. 2: Volatility, predictability (top) and revenue from s

**Figure 3.** FIG. 3: Effect of introducing a tax on the exchange rate fluc

**Figure 4.** FIG. 4: Volatility (top), predictability (middle) and reve

**Figure 6.** FIG. 6: Volatility as a function of

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603134 by the authors.

References

Works cited (25)

[1]

Tobin J, “The New Economics One Decade Older” , The Eliot Janeway Lectures on Historical Economics in Hon- our of Joseph Schumpeter 1972 (Princeton University Press, Princeton, US)T

View on Pith 1972
[2]

Tobin J, Eastern Economic Journal IV 153-159 (1978)

View on Pith 1978
[3]

Ramonet I, Le Monde Diplomatique “Disarming the mar- kets” editorial 1997

View on Pith 1997
[4]

Ehrenstein G, Int. J. Mod. Phys. C, 13, 1323-1331 (2003)

View on Pith 2003
[5]

FBE Letter A tax on foreign exchange transactions: a false solution to the challenges posed by ﬁnancial markets , March 2001

View on Pith 2001
[6]

Ehrenstein G, Westerhoﬀ F and Stauﬀer D Quantitative Finance, 5, 213-218, (2005). cond-mat/0311581

View on Pith Pith review arXiv 2005
[7]

F. Westerhoﬀ, J. Evol. Ec., 13, 53-70 (2003)

View on Pith 2003
[8]

R. Cont, Issues, Quantitative Finance Vol. 1, 223-236 (2001)

View on Pith 2001
[9]

R. Cont, J.-P. Bouchaud, Macroecon. Dyn. 4, 170 (2000)

View on Pith 2000
[10]

C. H. Hommes, Heterogeneous Agent Models in Eco- nomics and Finance , Tinbergen Institute Discussion Pa- pers 05-056/1, Tinbergen Institute (2005)

View on Pith 2005
[11]

Challet D, Marsili M and Zhang Y-C 2005 Minority Games (Oxford University Press, Oxford UK)

View on Pith 2005
[12]

Coolen A C C 2005 The Mathematical Theory of Minority Games (Oxford University Press, Oxford UK)

View on Pith 2005
[13]

Johnson NF, Jeﬀeries P and Hui PM 2003 Financial mar- ket complexity (Oxford University Press, Oxford UK)

View on Pith 2003
[14]

the simulations then deviate systematically from the theory at large ns when the system has entered the zone near the phase transition line. More precisely, one ﬁnds a threshold value ns(L) > n ⋆ s so that numerical simula- tions agree with the theoretical lines for ns < ns(L) but deviations and anomalous ﬂuctuations are observed for ns > ns(L). As L is i...

View on Pith 2002
[15]

Challet D and Marsili M Phys. Rev. E 68 036132 (2003)

View on Pith 2003
[16]

This is an assumption on the cognitive limitation of agents because the game itself will generate a much richer information than just the sequence µ(t)

View on Pith
[17]

This contrasts with the deﬁnition of µ(t) in the original MG, where the information was endogenously generated by the market, with µ(t) encoding the sign of the past M = log 2 P price changes. Most collective properties of the model have been shown to be aﬀected only weakly by the origin of information. We focus here on the simpler case of exogenous infor...

View on Pith
[18]

M. Marsili, Physica A 299, 93-103 (2001)

View on Pith 2001
[19]

We here assume that each agent holds only one trading strategy. Generalizations to multiple strategies per play er are straightforward, and can be seen not to alter the qual- itative behaviour of the model. Hence we here restrict to the simplest case

View on Pith
[20]

D. Challet, A. De Martino, M. Marsili and I. Perez Castillo,JSTAT P04002 (2004)

View on Pith 2004
[21]

M. Marsili, D. Challet, Phys. Rev. E 64 (5), 056138 (2001)

View on Pith 2001
[22]

J.A.F. Heimel, A.C.C. Coolen, Phys. Rev. E 63 056121 (2001)

View on Pith 2001
[23]

D. Challet private communication

View on Pith
[24]

Mezard M, Parisi G and Virasoro M Spin glass theory and beyond World Scientiﬁc Singapore (1987). 8 Appendix We here sketch the theoretical analysis of the model with τ = 0 and general values of ε. The introduction of a tax τ can be accounted for by replacing ε → ε + τ in all equations below. The starting point of the statistical mechanics approach is the ...

View on Pith 1987
[25]

− erf[(1 + K)ζ/ √ 2]) (12) . These results are fully exact in the thermodynamic limit, with no approximations (except for the replica- 9 symmetric ansatz) made at any stage. Finally, neglect- ing certain dynamical correlations between agents the volatility can be approximated as σ2 = ε2 np + nsQ(ζ, K ) (ns + np)K 2 + ns φ − Q(ζ, K ) ns + np . (13) As show...

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Tobin J, “The New Economics One Decade Older” , The Eliot Janeway Lectures on Historical Economics in Hon- our of Joseph Schumpeter 1972 (Princeton University Press, Princeton, US)T

View on Pith 1972

[2] [2]

Tobin J, Eastern Economic Journal IV 153-159 (1978)

View on Pith 1978

[3] [3]

Ramonet I, Le Monde Diplomatique “Disarming the mar- kets” editorial 1997

View on Pith 1997

[4] [4]

Ehrenstein G, Int. J. Mod. Phys. C, 13, 1323-1331 (2003)

View on Pith 2003

[5] [5]

FBE Letter A tax on foreign exchange transactions: a false solution to the challenges posed by ﬁnancial markets , March 2001

View on Pith 2001

[6] [6]

Ehrenstein G, Westerhoﬀ F and Stauﬀer D Quantitative Finance, 5, 213-218, (2005). cond-mat/0311581

View on Pith Pith review arXiv 2005

[7] [7]

F. Westerhoﬀ, J. Evol. Ec., 13, 53-70 (2003)

View on Pith 2003

[8] [8]

R. Cont, Issues, Quantitative Finance Vol. 1, 223-236 (2001)

View on Pith 2001

[9] [9]

R. Cont, J.-P. Bouchaud, Macroecon. Dyn. 4, 170 (2000)

View on Pith 2000

[10] [10]

C. H. Hommes, Heterogeneous Agent Models in Eco- nomics and Finance , Tinbergen Institute Discussion Pa- pers 05-056/1, Tinbergen Institute (2005)

View on Pith 2005

[11] [11]

Challet D, Marsili M and Zhang Y-C 2005 Minority Games (Oxford University Press, Oxford UK)

View on Pith 2005

[12] [12]

Coolen A C C 2005 The Mathematical Theory of Minority Games (Oxford University Press, Oxford UK)

View on Pith 2005

[13] [13]

Johnson NF, Jeﬀeries P and Hui PM 2003 Financial mar- ket complexity (Oxford University Press, Oxford UK)

View on Pith 2003

[14] [14]

the simulations then deviate systematically from the theory at large ns when the system has entered the zone near the phase transition line. More precisely, one ﬁnds a threshold value ns(L) > n ⋆ s so that numerical simula- tions agree with the theoretical lines for ns < ns(L) but deviations and anomalous ﬂuctuations are observed for ns > ns(L). As L is i...

View on Pith 2002

[15] [15]

Challet D and Marsili M Phys. Rev. E 68 036132 (2003)

View on Pith 2003

[16] [16]

This is an assumption on the cognitive limitation of agents because the game itself will generate a much richer information than just the sequence µ(t)

View on Pith

[17] [17]

This contrasts with the deﬁnition of µ(t) in the original MG, where the information was endogenously generated by the market, with µ(t) encoding the sign of the past M = log 2 P price changes. Most collective properties of the model have been shown to be aﬀected only weakly by the origin of information. We focus here on the simpler case of exogenous infor...

View on Pith

[18] [18]

M. Marsili, Physica A 299, 93-103 (2001)

View on Pith 2001

[19] [19]

We here assume that each agent holds only one trading strategy. Generalizations to multiple strategies per play er are straightforward, and can be seen not to alter the qual- itative behaviour of the model. Hence we here restrict to the simplest case

View on Pith

[20] [20]

D. Challet, A. De Martino, M. Marsili and I. Perez Castillo,JSTAT P04002 (2004)

View on Pith 2004

[21] [21]

M. Marsili, D. Challet, Phys. Rev. E 64 (5), 056138 (2001)

View on Pith 2001

[22] [22]

J.A.F. Heimel, A.C.C. Coolen, Phys. Rev. E 63 056121 (2001)

View on Pith 2001

[23] [23]

D. Challet private communication

View on Pith

[24] [24]

Mezard M, Parisi G and Virasoro M Spin glass theory and beyond World Scientiﬁc Singapore (1987). 8 Appendix We here sketch the theoretical analysis of the model with τ = 0 and general values of ε. The introduction of a tax τ can be accounted for by replacing ε → ε + τ in all equations below. The starting point of the statistical mechanics approach is the ...

View on Pith 1987

[25] [25]

− erf[(1 + K)ζ/ √ 2]) (12) . These results are fully exact in the thermodynamic limit, with no approximations (except for the replica- 9 symmetric ansatz) made at any stage. Finally, neglect- ing certain dynamical correlations between agents the volatility can be approximated as σ2 = ε2 np + nsQ(ζ, K ) (ns + np)K 2 + ns φ − Q(ζ, K ) ns + np . (13) As show...

View on Pith