Game theory and speculation on government bondsDavid Carfìa ,Francesco Musolino b ,⁎a Department of Mathematics,University of California at Riverside 900Big Springs Road,Surge 231Riverside CA 92521‐0135,USA bUniversity of Messina,Piazza Pugliatti,98122Messina,Italya b s t r a c ta r t i c l e i n f o Article history:Accepted 17June 2012Keywords:Financial Markets Financial Risk Financial Crises Game Theory SpeculationGovernment BondsThe aim of this paper is to propose a method to stabilize the rapid variations on the value of government bonds issued by the States,using Game Theory.In particular,we focus our attention on three players:a large speculative bank (hereinafter called Speculator),having immediate access to the market of government bonds,the European Central Bank (ECB)and a State in economic crisis,with a high public debt.In this regard,we will analyze the in-teraction between these three subjects:the Speculator,our first player,the ECB,our second player,and the State,our third player.The financial crisis,that hit the market of European government bonds,showed us that large speculators can in fluence the financial markets and bene fit from the creation of arbitrage opportunities caused by themselves.In this way,the default probability of States in economic dif ficulty increases signi ficantly and alarmingly.We already heard to talk about concepts like “spread ”and “public debt,”which has crippled the econ-omies of great States,for instance Italy.In this paper we propose on financial transactions the introduction of a tax,which hits only the speculative pro fits.We show how the above tax would probably be able to avert the speculation.For this purpose,we compare the different behaviors adopted by the Speculator and by the ECB in case of absence or presence of the tax,with the consequent effects on the State that sells its government bonds,paying particular attention to the movement of the game equilibria.In fact,with the introduction of our tax,all equilibria of the game become excellent for the State in economic dif ficulty.©2012Elsevier B.V.All rights reserved.1.IntroductionLately,the global economic crisis is increased,affecting even States considered very important in the economy (as for example Italy).One of the causes of the crisis is the exponential growth in government bonds yields,which have increased the public debt (up to May 2011the Italian government bonds offered a yield of approximately 4.80%,while in December 2011it rose above the 7.50%).Fig.1,made by Richard Portes,Professor of Economics at London Business School,shows the Ital-ian situation (see /news/in-pictures-16090055).In this regard,with our model (for a complete study of a game see Agreste et al.,2012;Baglieri et al.,2012;Carfì,2008,2009a,b,c,2010;Carfìand Musolino,2011a,b,2012a,b,c,d,submitted for publication-a,submitted for publication-b,submitted for publication-c;Carfìand Ricciardello 2010,2012a,b;Carfìand Schilirò,2012a,b;Musolino,2012,submitted for publication ),we intend to propose a possible meth-od to stabilize the government bonds markets of the States in economic dif ficulty,without any losses of collective gain.In this way,with the in-troduction of a simple but effective tax,the market would be able by it-self to reduce yields on government bonds,without further economic measures at global level:thus the States in financial dif ficulty could fi-nally begin (hopefully)a slow but steady economic recovery.2.MethodologiesThe normal-form game G ,that we propose to model our financial interaction,requires a construction which takes place 3times,which we say time 0,time 1/2and time 1.•At time 0the Speculator (the first player)can decide:1)to sell short government bonds,in order to obtain greater pro fitbetting on a greater future yield of the bonds;2)not to intervene in the government bonds market.•At time 1/2the ECB may decide to intervene in the bonds market in order to limit the growth of the bonds yield.In this way,even in case of lack of demand of government bonds,the issuer State finds the funds necessary to the national financial requirement.•At time 1the Speculator must eventually close its position (opened at time 0),by buying government bonds.Remark.During the game,we will refer to an interest rate i which de-termines the yield on government bonds.When we pass from one time period to another one,we should actualize or capitalize the values that must be “transferred ”.But because the interest rate i u ,usually adopted in the capitalizations and discounts in our financial markets,is much lower than that one we use to get the yield of government bonds,we as-sume i u equal to 0.Therefore,in this model,the values referred to differ-ent time period are not capitalized or discounted.Economic Modelling 29(2012)2417–2426⁎Corresponding author.E-mail addresses:davidcar fi71@yahoo.it (D.Carfì),francescomusolino@hotmail.it (F.Musolino).0264-9993/$–see front matter ©2012Elsevier B.V.All rights reserved.doi:10.1016/j.econmod.2012.06.037Contents lists available at SciVerse ScienceDirectEconomic Modellingj o u r n a l h o m e p a ge :w w w.e l s e v i e r.c o m/l o c a t e /e c m o d3.Financial preliminariesHere we recall the financial concepts that we shall use in the present article.1.M represents the quantity expressed in money of issued bonds (for example Italy has to issue a quantity equal to M of government bonds in order to face its financial commitments).2.Short selling of bonds is a financial transaction involving the sale of bonds without having their property,hoping to buy them later at a lower price.So the short seller would realize a pro fit.In the event that we examine,talking about government bonds,the hope of short sellers consists in an increase of the yield on government bonds.3.The government bonds are not normal goods with a purchase price and a sale price.The concept that characterizes them is the yield,which depends upon the interest to which they are sold.The yield on a government bond is given by the interest that remunerates the capital “loaned ”to the State.4.The description of the gameOur first player,the Speculator,may choose to sell short government bonds,in order to cause an increase in their yield and so to obtain a prof-it.In fact,at time 1the Speculator must close the position opened in the government bonds market with a purchase transaction.Otherwise the Speculator can decide not to intervene in the government bonds market.Thus,the Speculator has the possibility to choose among the strategies x ∈[0,1]which represents the percentage of the quantity of government bonds M that the Speculator decides to buy,depending if it intends:not to make any financial transaction (x =0);to sell short government bonds (xM is the quantity of short sold bonds)(0b x ≤1).On the other hand,the European Central Bank,that is our second player,operates in the bonds market in consequence of the operation of the first player.It may choose a strategy y ∈[0,1],which represents the percentage of the quantity of government bonds M that the ECB purchases,depending if it intends:to buy government bonds of the State in economic dif ficulty (y >0);not to intervene in the govern-ment bonds market (y =0).In Fig.2we illustrate graphically the bi-strategy space E ×F of our game G .5.The no tax game5.1.The payoff function of the Speculator in the no tax gameThe payoff function of the Speculator,which is the function that represents the gain of the first player,is given by the quantity expressed in money of purchased bonds xM ,multiplied by the differ-ence R 1(x ,y )−R 0between the value at time 1of the yield to be cashed (at time 1the Speculator buys the same amount of securities that it has sold short at time 0)and the value at time 0of the yield to be paid (at time 0the Speculator sells short a certain amount of government bonds).The payoff function of the Speculator is given by:f 1x ;y ðÞ¼xM R 1x ;y ðÞ−R 0ðÞ;ð1Þwhere:1)xM is the amount of government bonds that the Speculator shortsells at time 0;2)R 1(x ,y )is the value of the government bonds yield at time 1.Wesuppose it is given byR 1x ;y ðÞ¼i þmx −ny ;where:•i is the interest that remunerates the capital “loaned ”to the State;•m is a marginal coef ficient which indicates the incidence of x on R 1(x ,y );•n is a marginal coef ficient which indicates the incidence of y on R 1(x ,y ).The government bonds yield R 1(x ,y )depends on x because if the Spec-ulator intervenes in the government bonds market with a strategy x ≠0,the yield R 1(x ,y )is modi fied because a decline in demand has a positive effect on the interest charged on the government bond.R 1(x ,y )depends on y because if the ECB intervenes in the government bonds market with a strategy y ≠0,the value R 1(x ,y )is modi fied because an increase in demand has a negative effect on the government bonds yield (the interest that remunerates the bond goes down).We are assuming by hypothesis both for x and y a linear dependence.3)R 0is the value of the government bonds yield at time 0.It is givenby R 0=i ,where i is the interest remunerating the capital that is “loaned ”to the State.R 0is a constant because our strategies x and y have not impact onit.Fig.1.The yield on Italian government bonds in2011.Fig.2.The bi-strategy space of the game.2418 D.Carfì,F.Musolino /Economic Modelling 29(2012)2417–24265.1.1.The payoff function of the Speculator in the no tax gameTherefore,recalling the function R 1,the de finition of R 0and the function f 1,we have f 1x ;y ðÞ¼xM mx −ny ðÞ:ð2Þ5.2.The payoff function of the ECB in the no tax gameThe payoff function of the ECB,that is the function representing the algebraic gain of the ECB,is given by the multiplication of the quantity expressed in money of government bonds yM (which the ECB buys at time 1/2)by the bonds yield at time 1/2,that is R 1=2x ðÞ¼i þmx :So the payoff function of the ECB is given by:f 2x ;y ðÞ¼yMR 1=2x ðÞ;ð3Þwhere:1)yM is the quantity of bonds expressed in money that the ECB buysat time 1/2;2)R 1/2(x )is the bonds yield at time 1/2.It is given by:R 1=2x ðÞ¼i þmx :The strategy x has impact on the government bonds yield at time 1/2because at time 0the Speculator has already operated in the mar-ket,changing the bonds yield.5.2.1.The payoff function of the ECBRecalling functions R 1/2and f 2,we have f 2x ;y ðÞ¼yM i þmx ðÞ:ð4Þ5.3.The payoff function of the StateIn addition to the payoff functions of the Speculator and the ECB must also be considered the payoff function of the State.It is given by the quantity M of issued government bonds,multiplied by the differ-ence between the yield R 0(which the State would pay without the in-tervention on the market of the Speculator and of the ECB)and the yield R 1(which actually pays in consequence of the strategies x of the Speculator and y of the ECB).5.3.1.Payoff function of the StateThe payoff function f (3)of the State is given by:f 3x ;y ðÞ¼M R 0−R 1x ;y ðÞðÞ:ð5ÞRecalling the de finition of R 0,the function R 1,and the function f 3,we havef 3x ;y ðÞ¼M −mx þny ðÞ:ð6ÞThe payoff function of the no tax game is so given,for every (x ,y )∈E ×F ,by:f x ;y ðÞ¼xM mx −ny ðÞ;yM i þmx ðÞ;M −mx þny ðÞðÞð7Þ6.Study of the no tax game 6.1.Critical space of the no tax gameSince we are dealing with a non-linear game,it is necessary to study in the bi-win space also the points of the critical zone that belong to the bi-strategy space.In order to find the critical area of the game,we con-sider the Jacobian matrix and we put its determinant equal 0.About the gradients of f 1and f 2,we have ∇f 1x ;y ðÞ¼M 2mx −ny ðÞ;−nxM ðÞ∇f 2x ;y ðÞ¼Mmy ;M i þmx ðÞðÞ:Let g be the vector function from the bi-strategy space into the Cartesian plane,having its components conciding with the first two components of f .The determinant of the Jacobian matrix of g is:det J g x ;y ðÞ¼M 22mx −ny ðÞi þmx ðÞþM 2mxny :Therefore,the critical space of the game is:Z g ¼x ;y ðÞ:M 22mx −ny ðÞi þmx ðÞþM 2mxny ¼0n o:Dividing by M 2m ,which are all positive numbers (strictly greater than 0),after calculations finally we have:Z g ¼x ;y ðÞ:y ¼2mx i −mx ðÞni ðÞ−2f g :Assuming that m =1/2,n =1/2and i =1/4,we obtain Z g ¼x ;y ðÞ:y ¼1=2ðÞx 2þ1=4ðÞx n o:The critical area of our bi-strategy space is represented in Fig.3by the segment [D ,H ].6.2.Payoff space of the no tax gameIn order to represent graphically the payoff space g (E ×F ),we transform,by the function g ,all the sides of bi-strategy square E ×F and the critical space Z of the game (g,>).Fig.3.The critical zone of the no tax game.2419D.Carfì,F.Musolino /Economic Modelling 29(2012)2417–2426The segment[B,C]is the set of all the bi-strategies(x,y)such that x=1and y∈[0,1].Calculating the image of the generic point(1,y), we have:g(y,1)=(M(m−ny),yM(i+m)).Therefore,setting:X¼M m−nyðÞand Y¼yM iþmðÞ;and assuming M=1,i=1/4and n=m=1/2,we haveX¼1=2−1=2ðÞy and Y¼3=4ðÞy:Replacing Y instead of y in thefirst equation,we obtain the image of the segment[B,C],defined as the set of the bi-wins(X,Y)such that X¼1=2−2=3ðÞY with Y∈0;3=4½ :It is a line segment with extremes B′=g(B)and C′=g(C).Following the procedure described above for the other sides of the bi-strategy square and for the critical space,which are the seg-ments[A,B],[C,D],[D,A]and[D,H],we obtain in Fig.4the payoff space g(E×F)of our game(g,>).We note that we get a sail-formedfigure,but the results must now be interpreted according to the payoff function of the issuer State.Recalling the function f3,and that M=1and n=m=1/2,we note that:•if the two players arrive on the points D′and B′the yield(that the issuer State must pay for its government bonds)remains unchanged because it is balanced by two equal opposing forces.This solution is un-desirable because it does not solve the problems of the State and does not give breath to its economy;•if the two players arrive on the side]B′,C′],the yield paid by the State for its government bonds increases inexorably,bringing it closer to default;•if the two players arrive in[A′,D′[,the yield paid by the State for its government bonds decreases,and thus the State could emerge from the crisis.According to these considerations,it is morally,ethically and eco-nomically desirable that the Speculator and the ECB arrive to the point A=(0,1),so that the paid yield goes down as more is possible and the State comes out of the crisis.Remark.The point A′and the point B′have the same collective gain about the three subjects of our game.In fact,if we arrive to point A′the State in economic difficulty has a profit equal to1/2,the Specula-tor wins0and the ECB wins1/4.On the contrary,if we arrive to point B′the State has a profit equal to0,the Speculator wins0and the ECB wins3/4.In both points,the total gain of the game is3/4.7.Equilibria of the no tax game7.1.Nash equilibria of the no tax gameIf the two players decide to adopt a selfish behavior,they choose their own strategy maximizing their partial gain.In this case,we should consider the classic Nash best reply correspondences.The best reply cor-respondence of the Speculator is the correspondence B1:F→E given by y↦max f1(⋅,y)E,where max f1(⋅,y)E is the set of all strategies in E which max-imize the section f1(⋅,y).Symmetrically,the best reply correspondence B2:E→F of the ECB is given by x↦max f2(x,⋅)F.Choosing M=1,n=1/2 and m=1/2,which are always positive numbers(strictly greater than 0),and recalling that f1(x,y)=xM(mx−ny),we have∂1f1(x,y)= 2mxM−nyM.We haveB1yðÞ¼1f g if y b10;1f g if y¼1:Recalling also that f2(x,y)=yM(i+mx),we have∂2f2(x,y)= M(i+mx)and so:B2xðÞ¼1f g;∀x∈E:In Fig.5we have in red the inverse graph of B1,and in blue the graph of B2.The set of Nash equilibria,that is the intersection of the two best reply graphs(graph of B2and the symmetric of B1),is Eq.B1;B2ðÞ¼1;1ðÞ;0;1ðÞf g:7.1.1.Analysis of Nash equilibriaThe Nash equilibrium B=(1,1)can be considered very good for the two players,because they are on the proper maximal Pareto boundary.It is clear that if the two players pursue as aim the profit, and decide to choose their selfish strategy to obtain the maximum possible win,they will arrive on the proper maximal Pareto bound-ary.The selfishness,in this case,pays well.But the Nash equilibrium B=(1,1)does not solve the problems of the State that issues the gov-ernment bonds,because it should pay a yield that the strategy x=1 increases,and that is returned to its original level by the strategy y=1.In a word,the State continues to fund its public spending with government bonds,yielding too high for its possibilities.In the long term the State will end on the brink of the abyss.The Nash equilibrium A=(0,1),instead,is good for the State,be-cause the yield to pay on government bonds goes downward.But the point A′is not on the maximal Paretoboundary.Fig.4.The payoff space of the no taxgame.Fig.5.The Nash equilibria of the no tax game. 2420 D.Carfì,F.Musolino/Economic Modelling29(2012)2417–2426Moreover,most likely,the Speculator will choose the strategy x=1, because the strategy x=0precludes the opportunity of profit for the Speculator,which is stuck on the ordinate axis.With x=1,instead, the Speculator tries to win something depending on the strategy of the ECB,and still manages to not lose.Basically,the most likely Nash equilibrium is the point B=(1,1):almost certainly the achievement of a Nash equilibrium would leave the issuer State in trouble and at risk of default.Remark.At this point,the ECB could consider splitting the win3/4 obtained in the most likely Nash equilibrium B with the issuer State, in order to cancel the effects of the increase of the yield on govern-ment bonds.Thus,the ECB would give1/2(value that the State loses because of a strategy x=1)to the State,taking for itself the sum of1/4.But this seemingly simple solution is not feasible for sev-eral reasons:1.the ECB has a policy that usually does not interfere with that one ofthe European States,therefore this kind of action is difficult to accomplish;2.the“payback”to the State by the ECB could have very long timescales,and therefore the State could sink even deeper into economic crisis.3.the amount cashed by the State cancels its loss(suffered because ofthe strategy of Speculator),but the yield on government bonds would remain high,and then this action is not a solution:it would simply postpone the problem over time without dealing with it.In fact,if in the future otherfinancial institutions buy(also without speculative purposes)government bonds,the State should pay them a yield which is remained at unsustainable levels,ending in bankruptcy.For these reasons,it is important tofind a method that allows the State to prevent speculation and not to be constantly“cured”.Anyway, it is obvious that a vaccine made only once is better than a medicine taken continuously,a medicine which in the future will lose its effectiveness.Note.We can note that there are three possible cases:1.If m=n we have the case that we are studying.2.If m>n we have:B1(y)={1}∀y∈F.3.If m b n we have:B1yðÞ¼1f g if y b m=n0;1f g if y¼m=n0f g if m=n b y≤1: 8<:In order that our model achieves the aims which will be explained fur-ther on,it is necessary that the marginal incidence m of the strategy x on R1is not lower than the marginal incidence n of the strategy y on R1(in fact in this case the point A=(0,1)is already the only Nash equilibrium). But very likely,the value n is lower than the value m because the purchase of government bonds by the ECB could be less accepted by the market players.In fact,the action of the ECB could be seen as a behavior dictated (also,or even only)by political motivations,and not by economic reasons (like for example low risk and high profit about government bonds).7.2.Defensive phase of the no tax gameWe suppose that the two players are aware of the will of the other one to destroy it economically,or are by their nature cautious,fearful,para-noid,pessimistic or risk averse,and then they choose the strategy that al-lows them to minimize their loss.In this case,we talk about defensive strategies.7.2.1.Conservative value and crosses7.2.1.1.Conservative value of a player.It is defined as the maximization of its function of worst win.Therefore,the conservative value of the Speculator is v1#=sup x∈E f1#(x),where f1#is the function of worst win of the Speculator,and it is given by f1#(x)=inf y∈F f1(x,y),for every x in E.Choosing M=1,n=0.5and m=0.5,which are always positive numbers(strictly greater than0),and recalling that f1(x,y)= xM(mx−ny),we have:f1#(x)=inf y∈F xM(mx−ny).Since the offensive strategies of the ECB are O2xðÞ¼1f g if x>0F f g if x¼0;we obtain:f1xðÞ¼xM mx−nðÞf g if x>00f g if x¼0:In Fig.6appears f1#graphically.So the defense(or conservative)strategy set of the Speculator is E#={0,1}and the conservative value of the Speculator isv#1¼supx∈Einfy∈FxM mx−nyðÞ¼0:ð8ÞOn the other hand,the conservative value of the ECB is given by v2#=sup y∈F f2#,where f2#is the function of the worst win of the ECB. It is given by f2#(y)=inf x∈E f2(x,y),for every y∈F.Choosing M=1, i=0.25and m=0.5,which are always positive numbers(strictly greater than0),and recalling that f2(x,y)=yM(i+mx),we have: f2#(y)=inf x∈E yM(i+mx).Since the worst offensive strategies of the Speculator are O1yðÞ¼0f g if y>0E f g if y¼0,we obtain:f2yðÞ¼yMif g if y>00f g if y¼0:In Fig.7appears f2#graphically.Fig.6.The function of worst win of the Speculator in the no taxgame.Fig.7.The function of worst win of the ECB in the no tax game.2421D.Carfì,F.Musolino/Economic Modelling29(2012)2417–2426So the defense(or conservative)strategy set of the ECB is given by F#={1}and the conservative value of the ECB isv#2¼supy∈F infx∈EyM iþmxðÞ¼Mi:ð9ÞTherefore,recalling that M=1and i=0.25,the conservative bi-value isv# f ¼v#1;v#2¼0;1=4ðÞ:7.2.2.Conservative crossesThey are bi-strategies(x#,y#),whose coordinates are conservative strategies of thefirst and second player respectively.We have only two conservative crosses:the points B=(1,1)and A=(0,1).The conserva-tive cross B=(1,1)can be considered good for the Speculator and the ECB,because it is on the maximal Pareto boundary,but it is mediocre for the State.In fact,recalling that f3(x,y)=M(−ny+mx),the yield to pay on government bonds goes down by1/2,in accordance with the strategy y=1of the ECB,but it re-increases by the same amount because of the strategy x=1of the Speculator.The conservative cross A=(0,1),instead,is good for the State,be-cause the yield to pay on government bonds goes downward.But the point A′is not on the maximal Pareto boundary.Moreover,most likely, the Speculator will choose the strategy x=1,because the strategy x= 0precludes the opportunity for profit for the Speculator,which is stuck on the ordinate axis.With x=1,instead,the Speculator tries to win something depending on the strategy of the ECB,and still manages to not lose.Basically,the most likely conservative cross is B=(1,1):al-most certainly the achievement of a conservative cross would leave the issuer State in trouble and at risk default.Note.Recalling that f#1xðÞ¼xM mx−nðÞf g if x>00f g if x¼0we can notethat there are three possible cases:1.If m=n we have the case that we are studying.2.If m>n we have:x#=1.3.If m b n we have:x#=0.In order that our model achieves the aims which will be explained further on,it is necessary that the marginal incidence m of the strategy x on R1is not lower than the marginal incidence n of the strategy y on R1 (in fact in this case the point A=(0,1)is already the only defensive equilibrium).But very likely,the value n is lower than the value m be-cause the purchase of government bonds by the ECB could be less ac-cepted by the market players.In fact,the action of the ECB could be seen as a behavior dictated(also,or even only)by political motivations, and not by economic reasons(such as for example low risk and high profit about government bonds).7.3.Cooperative solutions of the no tax gameThe best way for the two players to get both a win without causing the default of the State in economic difficulty,is tofind a cooperative solution.7.3.1.Cooperative solutionThe Speculator and the ECB play the strategies x=0and y=1in order to arrive at the payoff A′(which allows the State to reduce the yield on its government bonds)and then they split the bi-win A′by means of a contract.The Speculator benefits by cooperating with the ECB because following the Nash strategy it does not win anything (while in this way wins1/10);the ECB is able to save the State in diffi-culty,but gives up a significant part of its win than the Nash equilibrium.Practically,the Speculator does not act with any speculative move-ment on the securities market,and the ECB,that manages to save the State in economic crisis,shares with the Speculator its winning W=1/4, obtained arriving to A′.For a possible quantitative division of this win W=1/4,between the ECB and the Speculator,we use the transferable utility solution, applying to the transferable utility Pareto boundary of the payoff space a non-standard Kalai–Smorodinsky solution.Remark.We consider the infimum and the supremum of the maxi-mal Pareto boundary for a better view of the game in its entirety.We proceedfinding the supremum of our maximal Pareto bound-ary,which is sup∂∗g(E×F)=:α=(1/2,3/4);then we join it with the infimum of our maximal Pareto boundary,which is given by inf∂∗g(E×F)=(0,0).The coordinates of the intersection of the point P,between the straight line of collective win(i.e.X+Y=1/4)and the straight line joininig the supremum of the maximal Pareto boundary with the infimum(i.e.the line Y=(3/2)X),give us the desirable division of the collective win W=1/4between the two players.In order tofind the coordinates of the point P is enough to put in a system of equations X+Y=1/4and Y=(3/2)X.Substituting the Y in thefirst equation we have X+(3/2)X=1/4and therefore X=1/10. Substituting now the X in the second equation,we have Y=3/20. Thus P=(1/10,3/20)suggests as solution that the Speculator re-ceives1/10by contract by the ECB,while at the ECB remains the win3/20.We can see Fig.8in order to make us more aware of the situation. Remark.But the cooperative solution leaves us dissatisfied.In fact the cooperative solution is difficult to implement because the ECB should achieve an agreement with the Speculator before that the Speculator plays a strategy x>0,and is almost impossible to know in advance the intentions of all the potential speculators in the gov-ernment bonds market.For this reason,it is necessary to have a pre-ventive economic measure.8.Payoff functions of the game with tax8.1.The payoff of the Speculator in the game with tax8.1.1.The anti-speculative proposalWe propose,in order to avoid speculations of thefirst player about the current and future yield of the government bonds,thatthe Fig.8.A possible cooperative solution of the no tax game.2422 D.Carfì,F.Musolino/Economic Modelling29(2012)2417–2426。
