Vadim V.Romanuke
Approximation of Unit-Hypercubic Infinite Noncooperative Game Via Dimension-Dependent Samplings and Reshaping the Player’s Payoffs into Line Array for the Finite Game Simplification
Vadim V.Romanuke1
The problem of solving in finite noncooperative games approximately is considered.The game may either have solution or have no solution.The existing solution may be unknown as well.Therefore,an approach of obtaining the approximate solution of the in finite noncooperative game on the unit hypercube is suggested.The unit-hypercubic game isomorphism to compact in finite noncooperative games allows to disseminate the approximation approach on a pretty wide class of noncooperative games.The approximation intention is in converting the infinite game into a finite one,whose solution methods are easier rather than solving in finite games.The conversion starts with sampling the players’payoff functions.Each dimension of the player’s pure strategies unit hypercube is sampled with its own sampling constant,being the number of equal-measure intervals between the selected points along the dimension.There are stated requirements for the sufficiently accurate sampling.Having got the finite game on hypercubic lattice after the sampling,every player’s payoff multidimensional matrix is reshaped to reduce number of its dimensions down to the number of players.Dimensionality reduction will commonly accelerate computations,connected with the approximate solution consistency.The introduced consistency mechanism rejects the finite game solution,pretended to being the initial game approximate solution,if the solution depends vastly on the sampling steps.If the solution is weakly consistent then,changing the sampling steps minimally,there are non-decreasing difference of the players’payoffs and difference of the players’equilibrium strategies and cardinalities of their supports.If the solution is consistent then the non-decreasing property holds stricter for cardinalities of the supports and their upper densities.
Infinite noncooperative game; Unit hypercube; Multidimensional matrix;Strategy support;Approximate solution consistency,Game isomorphism.
Any infiniteness in game modeling is hard and heavy to handle it.On the one hand,it is difficult to solve infinite noncooperative games analytically.On the other hand,often the infinite game solution appears impossible for its full practical realization,because an infinite support of the equilibrium strategy is practiced only over a zeromeasure subset of objects.Moreover,any of these objects(pure strategies)cannot be taken for infinite number of times.Therefore noncooperative game approxima-tion aims at two goals: to solve the game easier and to implement the solution as fast as possible[Osborne(2003)].And thus easy-and-fast solution is standing against the genuine solution.The matter is that some of important features of the genuine solution may be lost after approximation.It is crucial in rational resources allocation problems[G˛asior and Drwal(2013);Ye and Chen(2013)]with two sides of interest,military processes organization and jurisprudence,involving a few players[Suzdal(1976);Calvert,McCubbins,and Weingast(1989)],socio-economic and ecological gaming models with more players[Friedman(1998);Vorob’yov(1985)],multiagent modeling[Fujii,Yoshimura,and Seki(2010)],multiobjective optimization under uncertainty[Amaziane,Naji,Ouazar,and Cheng(2005);Li and Li(2010);Li,Li,Sun,Luo,and Zhang(2010);Li,Zhao,and Ni(2013);Zhu,Liu,Wang,and Yu(2004);Li,Luo,and Sun(2011);Trapani,Kipouros,and Savill(2012)],etc.So,there is needed a balanced way in searching easy-and-fast solution by approximating infinite noncooperative games.
Not every approximate solution of infinite noncooperative game implies the players’finite supports[Bernhard and Shinar(1990);Chakrabarti(1999)].Solving approximately can be either a method of thorough manipulation with infinite noncooperative game[Reny(1999);Mallozzi,Pusillo,and Tijs(2008)]or mapping this game into finite one[Zielonka(1998);McNaughton(1993)],whereupon it may be solved as approximately as well as exactly(analytically).Surely,getting the finite game is strongly preferable.However,there is no general theory of solving finite noncooperative games.It even is unclear what this theory could have been.Solving a noncooperative game always is springing up as an original problem,needing usually specif i c reasonings[Osborne(2003);Vorob’yov(1985);Kostreva and Kinard(1991);Pavel(2007)].Surely,these reasonings depend on the type of equilibrium,including utility,stability,or equity.Particularly,finding Nash equilibrium solutions in even the finite noncooperative game bears a computational difficulty[Osborne(2003);Vorob’yov(1985);Han,Zhang,Qian,and Xu(2012);Chen and Deng(2007)].Just when the player has minimal number of pure strategies,and there are no more than three players,the solution is analytical and there is a known technique of its visualization[Vorob’yov(1985);Vorob’yov(1984);Romanuke(2010)].Thus,solving dyadic games with three players is visualized on the cube of situations in mixed strategies,whereas dyadic games with four players and more are solved purely in analytics,requiring more computational resources[Vorob’yov(1984);Browning and Colman(2004)].But finite noncooperative games with greater numbers of pure strategies at their players(three and more)are much harder to solve them[Osborne(2003);Han,Zhang,Qian,and X-u(2012);Vorob’yov(1984);Nisan,Roughgarden,Tardos,and Vazirani(2007)].Nonetheless,difficulties in solving infinite noncooperative games are hardly comparable to those ones while a finite game is solved.
Compact games,having solutions at least in mixed strategies for measurable payoff functions[Osborne(2003);Vorob’yov(1985);Vorob’yov(1984);Kukushkin(2011);Stoltz and Lugosi(2007)],cannot be solved by a universal algorithmic approach,unless they are finite games.But compact games have ε-equilibrium situations with finite supports.Therefore they are reduced via constructing ε-nets[Vorob’yov(1984);Giannopoulos,Knauer,Wahlström,and Werner(2012)]for some ε> ε0,where ε0> 0 is a minimal distance between two strategies of the player by Helly metric[Vorob’yov(1984)].However,it is unknown how to select ε0.An overestimated value ε0drives to rough approximation,where some of important features of the genuine solution may be lost.And too small value ε0provokes either long-convergent method for approximate solution of the finite game or enormous computational spendings to get that finite game exact solution.
Let there be the noncooperative game withN∈N{1}players.And letn-th player act within the unitMn-dimensional hypercube
Thus pure strategy ofn-th player isMn-dimensional point
and in the situation
n-th player gets the payoffKn(X).Consequently,each of the measurable and bounded payoff functionsis defined on the unit-dimensional hypercube
And the tuple
is the noncooperative game on hypercube(3).This game is isomorphic[Osborne(2003);Vorob’yov(1985);Vorob’yov(1984)]to noncooperative games,defined on compact subspaces inwhereon payoff functions are measurable and bounded.The goal is to convert the infinite noncooperative game(4)on unit hypercube(3)into a finite game,whose formal representation shall be as simple as possible.The finite game solution of a certain type should reflect all significant features of the game(4)genuine solution.In this way the game(4)is going to be approximated,overlapping the easy-and-fast conception with solution genuineness.For that the players’payoff functions are going to be sampled finitely,with the pre-assigned constant sampling step in each ofdimensions,regarding special requirements to the sampling,which are to ensure sufficient accurateness for practice experience.Next task is to reconfigure the sets of the players’pure strategies in the finite game for its total Simplification,which will allow to get rid off dimensionalities and to have the single dimension at each player.Eventually,every player’s finite equilibrium strategy in the finite game as the approximation of the initial infinite game(4)mustn’t be too dependent upon the sampling constants.Hence requirements to every strategy support should be stated so that it would be independent upon the sampling constants within some tolerable dependence.
Following this,then-th player’s pure strategies set becomes
And so then-th player’s payoff function is sampled,f i xing its payoff values
and letting the game(4)be converted into the finite one
Note that the finite game(6)is defined on hypercubic lattice
now.
Clearly that constants
shouldn’t be assigned arbitrarily.They must be such that specif i cities of the players’payoff functionswould be preserved.These specif i cities consist mainly in local extremums and gradient over hypersurfacesSupposing that these hypersurfaces are differentiable with respect to any of variablesand there exist mixed derivatives of each of functionsby any combination of variablesin any situation(2),where every variable is included no more than just once,there are the following requirements.
But requirements(9)are obviously satisfied only if there are no extremums or discontinuities on any of intervals
So they can hardly be satisfied,unless there are two players and minimum of hypercube(3)dimensions.Far more real requirements are that on every of segments
fluctuations of the players’payoff functions would be no greater than some α > 0.Properly,there ought to be
Based on practical reasonings,
by,say,β =0.05,β =0.01,β =0.005 or β =0.001,what shall be sufficiently accurate for practice experience.However,the parameter α may be taken lesser to have the game approximate solution consistent enough,what is going to spoken about below.
In the finite game(6),then-th player’s payoff function,defined on hypercubic lattice(7),is represented as-dimensional matrix
with denotation
of the format
determine the matrix element In computations,there is a known rule telling that manipulating with a many single-dimensional objects is more convenient than manipulating with single multidimensional object[Hoffbeck,Sarwar,and Rix(2001);Rahman and Valdman(2013);Trapani,Kipouros,and Savill(2012)].Practically it is explained with that the greater supplementary dimensions of a matrix the longer computations might be.This computational retardation is easy exampled in Matlab environment.Suppose that a 12-dimensional4-matrix represents the player’s payoff values in 3-person game(each player’s pure strategy is of four dimensions with four points in every dimension).While operating on AMD Athlon II X2 250u Processor with 2 GB RAM within64-bit Windows7,1400 Matlab operations of summing and extracting mean,finding minimal and maximal elements of three such matrices take about 3756 seconds,whereas the same takes about 2919 seconds over three six-dimensional16-matrices,reshaped before.Moreover,reshaping these4-matrices into three-dimensional256-matrices reduces the operation time down to 2691 seconds.Therefore matricesshould be reshaped to reduce number of their dimensions ultimately.The minimal number of dimensions,apparently,is number of players.Here maintenance of one-to-one indexing is provided with the next theorem.
Theorem 1.There is a one-to-one indexing map of-matrix(11)into matrix of the format
This map is reversible.
Proof.Let-matrix(11)be reshaped into-matrix
whose elements
haveN-position indices,gathered within the set
Having denoted
the convolution mapping(17)shows thatReversely,let the function ψ(a,b)byb/=0 round the fractionto the nearest integer towards zero.And put another function
By the function(18),the last index in indicating then-th player’s aggregate indexunfor matrix(11)is
The restMn-1 indices are
Thus convolving statement(17)along with expansion(19)and(20)give the oneto-one reversible map of setinto setThe theorem has been proved.
Theorem 1 establishes correspondence of subset of indices
to a pure strategy ofn-th player and backwards,This simpli fies game(6)formalism ultimately,mapping the game
into
where then-th player’s pure strategycorresponds to its strategy Xnin the initial game(4),whose components are
due to subset(21)and indexuncorrespondence.
Without mentioning a method of solving the finite game(23),there is supposition of that the game(23)solution
And may the support of then-th player’s strategy(25)be the set
and
For seeing whether the strategy support(26)is independent upon the sampling constants(8)within some tolerable dependence,consider δ-neighboring to(23)game
with denotations
An aggregate feature of situation(28)or supportsis thei-th player’s payoff,being taken in this situation:
Apparently,there can be selected such constants(8),for which at leastsuch that payoffsandwill be significantly different.In other words,decreasing sampling steps minimally may give payoffsas if they do not relate to payoffsSimilarly,decreasing sampling steps minimally may give an equilibrium situation
whose configuration is hardly comparable to the corresponding configuration of situation(24).Therefore,approximating the unknown genuine equilibrium strategy requires two items.Firstly,payoffsandtaken in situations(24)and(30),should be sufficiently close in R-metric.Secondly,situations(24)and(30)themselves should be sufficiently close uniformly and in-metrics.
Of course,the spoken sufficient closeness is relative,meaning that the attribute value(every player’s payoff and its strategy)differentiates no greater as the numbers of the set(12)increase minimally(in comparison to that when they decrease minimally).Relativity is unavoidable because neither genuine payoffsin the game(4),nor limits
Sufficient closeness of the players’payoffs is that
sufficient closeness of situations,giving those payoffs in(32),needs consideration of the player’s strategy finite support as a hypersurface.Henceforward in the finite game(23),letn-th player have a piecewise linear hypersurface σn(un,S0),whose vertices are in points
is then-th player’s equilibrium strategy support in the game(6).And let the set(33)be sorted into the set
so that the value
is reached atq1=q+1 for eachandn=1,N.This is being made for evaluating sufficient closeness of finite strategies,when the sampling constants(8)vary minimally.Partially,this sufficient closeness is that
and
Fully,the upper support density shall also not decrease by the decreased sampling steps minimally:
The Definition below,engaged sufficient closeness in every player’s payoff and its strategy,is to see whether(24)is worth to count it the approximate solution of the game(4).
Definition 1.The solution(24)of the game(23)is called weakly consistent for being the approximate solution of the game(4)if the inequalities
are true along with(32)and(36)–(38).Every strategy and its support in weakly consistent solution are called weakly consistent.
Inequalities(39)express a natural requirement that cardinality of every player’s strategy support shall not decrease as the sampling steps by constants(8)decrease minimally.Weakness of consistency has been inserted inasmuch as requirements of the non-decreasing upper support density and support cardinality in(38)and(39)can be strengthened.
Definition 2.The weakly consistent solution(24)of the game(23)is called consistent for being the approximate solution of the game(4)if the inequalities
and
are true.Every strategy and its support in consistent solution are called consistent.Well,consistency or,accurately,S0-consistency subtends five stands,implicating the sets{S-1,S0,S1}.All issues from the player’s support:the support configuration,generating the payoffs in the equilibrium situation,the support cardinality,and the upper support density.Appositely,the support density is treated absolute,using Euclidean distance between points of the support.This has been acted for distinguishing the upper support density of the player’s completely mixed strategy as the sampling constants(8)vary.Hence the assertion below forces itself.
Theorem2.If weakly consistent strategy is completely mixed then it is consistent.
Proof.Let weakly consistent equilibrium strategy(25)ofn-th player be completely mixed.Then
Easily noting that
and
have
what conf i rms the inequality(40).Because strategy(25)is completely mixed,its support is the set(5).Consequently,indices
with their values
are such that there is the singlek0∈Bsuch that
and
and,subsequent upon(42),
Value(43)is clear to be minimal upper support density.According to this,
Once again easily noting that
have the inequality(41)confirmed.The theorem has been proved.
For the completely mixed strategy,inequalities(40)and(41)hold true strictly.Appositely,the case when inequalities(32)and(36)–(39)hold true strictly might have been called strict weak consistency,and strict consistency would have been on strict inequalities(32)and(36)–(41).That could be used in proving some limit theorems,but now questions of consistency computational approach are of higher importance.
Theorem 3.If the game
has a completely mixed equilibrium situation, then for checking weakS0-consistency of the same equilibrium type situation it is sufficient to check inequalities(32),(36),(37).
Proof.Apparently,
and
with applying(42)and(44)and(45)to the completely mixed strategies.So,inequalities(38)and(39)hold true,and there remain the inequalities(32),(36),(37)to be checked whether the game(23)solution(24)is weaklyS0-consistent.The theorem has been proved.
The proved assertions help either in reducing computations on consistency or rejecting the non-consistent solutions faster.Being valid on completely mixed strategies,they operate the support cardinality and upper support density.The investigator is brought to control inequalities(32),(36),(37),though.
This article concerns conversion of the unit-hypercubic infinite noncooperative game into a finite game.Validity of the conversion is grounded on sampling the players’payoff functions regularly,whereupon the obtained finite game solution is checked for its consistency.The equilibrium type is not specified nevertheless.It may be as Nash equilibrium type,as well as a lot of the refined or modified principles of optimality,allowing to smooth differences in utility and equity:strong Nash equilibrium[Suh(2001);Tian(2000)],Pareto equilibrium[G˛asior and Drwal(2013);Vorob’yov(1984,1985);Scalzo(2010);Zhu,Liu,Wang,and Yu(2004);Li,Luo,and Sun(2011);Trapani,Kipouros,and Savill(2012)],perfect Bayesian equilibrium[Fudenberg and Tirole(1991);Battigalli(1996)],Mertens-stable equilibrium[Kohlberg and Mertens(1986)],Markov perfect equilibrium[Castro and Brandão(2000);Haller and Lagunoff(2010)],etc.
The proposed approximation of the infinite noncooperative game allows to solve the finite game easier thanking to that every player’s payoff sampled function is reshaped into the line array,whereas computations over the multidimensional matrix with minimally possible number of dimensions are faster.Theorem 1 guarantees that the reshaping is the one-to-one indexing map,which is reversible.Reversibility is necessary in restoring the finite game(22)solution from the solution(24)of its simplified analogue(23).
Also the proposed approximation of the infinite noncooperative game calls for consistency of equilibrium strategy support,approximating the unknown genuine e-quilibrium strategy.Weak consistency by Definition 1 signifies that difference of the players’payoffs and difference of the players’equilibrium strategies and cardinalities of their supports are non-decreasing.This property becomes stronger by consistency in Definition 2,which imparts relative independence to the approximate solution upon the sampling steps within their minimal neighborhood in each of dimensions of hypercube(3).
In checking weakS0-consistency of the approximate solution,there are 5Ninequalities(32)and(36)–(39)to be checked.The check consecution starts with checking the inequalities(39),needing only solution of 1-neighboring to(23)game.If they all are true then inequalities(32)are checked,where(-1)-neighboring to(23)game is solved.If inequalities(32)are true then the support sufficient closeness in inequalities(36)and(37)is checked.And it is efficient that the problem(35)for sorting points(33)in the set(34)be solved after the sufficient closeness of equilibrium strategies is verified.Eventually,inequalities(38)are checked.CheckingS0-consistency should always follow the fact that the solution is weakly consistent.It starts with inequalities(40).If they all are true then inequalities(41),related to sorting problems,are checked.Namely the stated consecutions are preferable,because the easiest requirements are checked before the more complicated ones in order to prevent needless huge computations.
Deficiently,existence of limits(31)and their convergence to the genuine players’payoffs protrudes non-proved.Poorly that the limits
existence and their convergence to the being approximated equilibrium strategies have been left non-proved as well.Besides,it is unclear whether consistent strategy support causes at least the weak consistency of the other strategy support.Say,could a player’s strategy support be inconsistent while the rest ofN-1 players have their supports(weakly)consistent?Or else:shall a player use its(weakly)consistent strategy while the rest ofN-1 players have inconsistent supports?After all,the consistency paradigm can be regarded not only to equilibrium strategies but also to others(for instance,when a player decides to spring off an equilibrium situation).
In spite of everything,either conditions within Definition 1 or conditions within Definition 2 suggest a proper approximation of the infinite noncooperative game(4).The game(4)isomorphism to noncooperative games by measurable and bounded payoff functions,defined on compact subspaces inensure dissemination of that approximation approach on compact games.Further to this,the stated method of converting the unit-hypercubic infinite noncooperative game into the finite game lets have an equilibrium solution to the conflict object,even when
the game(4)is solved in ε-equilibrium situations or doesn’t have solution at all.And every player,having the finite strategy support,will practice it freer unlike practicing on infiniteness or with continuous variates.Computing the factual solution stays for finite noncooperative game solvers[Osborne(2003);Vorob’yov(1984);Nisan,Roughgarden,Tardos,and Vazirani(2007);Kuhn(1961);Kontogiannis,Panagopoulou,and Spirakis(2009)],wherein the computation period is shortened due to the minimized number of dimensions.
The game approximation is going to be brought forward:for each player,there will not be necessarily equal-measure intervals between the selected points in every dimension of the player’s pure strategies hypercube.This will let have irregular multidimensional hypercubic lattice instead of hypercube(1)wherewith to construct payoff matrices regarding straightforwardly any specif i cities,local extremums,and gradient over hypersurfacesgetting rid off requirements(9)and(10).Additionally,the statedS0-consistency,implicated the sets{S-1,S0,S1}or δ-neighboring to(23)games by δ ∈ {-1,0,1},is going to be extended out to δ-neighborhoods,considering wider symmetric ranges of δ.
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1Department of Applied Mathematics and Social Informatics,Khmelnitskiy National University,Khmelnitskiy,Ukraine
Computer Modeling In Engineering&Sciences2015年26期