2. Properties of the Information Market Game

Volume 2012, Article ID 379848,12pages doi:10.1155/2012/379848

Research Article

The Core and Nucleolus in a Model of Information Transferal

Dongshuang Hou and Theo Driessen

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Correspondence should be addressed to Dongshuang Hou,[email protected] Received 29 May 2012; Accepted 28 August 2012

Academic Editor: Marco H. Terra

Copyrightq2012 D. Hou and T. Driessen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Galdeano et al. introduced the so-called information market game involving n identical firms acquiring a new technology owned by an innovator. For this specific cooperative game, the nucleo- lus is determined through a characterization of the symmetrical part of the core. The nonemptiness of the symmetrical core is shown to be equivalent to one of each, super additivity, zero- monotonicity, or monotonicity.

1. Introduction of the Information Market Game

Consider the following problem1. Besidesnfirms with identical characteristics, there exists an agent called the innovator, having relevant information for the firms. The innovator is not going to use the information for himself, but this information can be sold to the firms. Any firm that decides to acquire the new information e.g., a new technologyis supposed to make use of the information. Thenpotential users of the information are the same before and after the innovator offers the new technology. The firms acquiring the information will be better than before obtaining it, while their utilities are computed under a conservator point of view, assuming that for any uninformed firm, the probability of making the right decision can be described by a binomial probability distribution, being 0≤p≤1 the uniform probability of having success. The probability thatk amongnfirms take the right decision is given by ⁿ_k·p^k·1−p^n−k, and hence, the expected aggregated utility ofkfirms having success is given byk·ⁿ_k·p^k·1−p^n−k·u_k. Hereu_k≥0 represents the utility ifkfirms make a right decision.

Throughout the paper, the utility function is monotonic decreasing because when the number of firms taking the right decision increases, each firm receives a lower utility level, that is, u_k1≤u_kfor allk≥1not necessarily normalized in thatu₁1.

This information trading problem has been modeled by Galdeano et al. 1 as a cooperative game N, v in characteristic function form, where the set of firms N {1,2, . . . , n1}consists of the innovator 1, having new information, and the users 2,3, . . . , n1, who could be willing to buy the new information. Throughout the paper, the size or cardinalityof any coalitionS⊆Nis denoted bys,0≤s≤n1. In case coalitionScontains the innovator, then its worthvSin the so-called information market game equalss−1·u_n because any member ofS, different from the innovator, took the right decision rewarding the expected utilityu_nsince then−suninformed firms outsideSare assumed to take right decisions too.

Definition 1.1. Then1-person information market gameN, vin characteristic function form is given byv∅ 0 and on the one handcf.1,

vS s−1·un ∀S⊆N with 1∈Sand on the other, 1.1 vS fns ^s

j1

j· s

j

·p^j·

1−p_s−j

·u_n−sj ∀S⊆N, S /∅,1∈/S. 1.2

If the innovator is not a member of coalitionS, each one ofksuccessful users rewards an expected utility the amount of^s_k·p^k·1−p^s−k·u_n−skby assumption of the uninformed users outsideStaking the right decisions. Particularly, the information market game satisfies v{1} 0, and v{i} f_n1 p·u_n for all i ∈ N,i /1. Furthermore,vN n·u_n, vN\ {i} n−1·unfor alli∈ N,i /1, whereasvN\ {1} fnn. Consequently, the marginal contributionsb_i^vvN−vN\ {i},i∈N, are given byb^v_i u_nfor alli∈N,i /1, whereasb^v₁ n·u_n−f_nn. It is left to the reader to verify

vN−vS

i∈N\S

vN−vN\ {i} ∀S⊆N with 1∈S. 1.3

The casep 1 yields vS s·u_n for all S ⊆ N\ {1}and so, it concerns the inessential additivegame corresponding with the vector0, un, un, . . . , un∈Rⁿ¹. The casep0 yields zero worth to all coalitions not containing the innovator and so, it concerns the so-called big boss game2 with the innovator acting as the big boss. We summarize the main results of Galdeano et al.1.

Theorem 1.2. For then1-person information market gameN, vof the form1.1-1.2, the following three statements are equivalent.

iZero-monotonicity, that is,

vS∪ {i}≥vS v{i} ∀i∈N, S⊆N\ {i}, 1.4

iis·u_n≥f_nsfor all 1≤s≤n, iii(cf. [1, Theorem 2, page 25])

un

u₁ ≥ p·

1−p_n−2 1p·

1−p_n−2 applied to the normalizationu₁1. 1.5

Besides their study of zero-monotonicity, Galdeano et al. determine the Shapley value of the information market gamecf. Theorem 4, page 27and compare the Shapley value with the equilibrium outcomecf. Theorem 7, page 29in the noncooperative model analyzed by3.

The main goal of the current paper is to determine the nucleolus of the information market game and for that purpose, we explore and characterize the symmetrical part of the core, provided nonemptiness of the core.

2. Properties of the Information Market Game

This section reports properties of the characteristic function for the information market game. In fact, we claim the equivalence of three game propertiescalled super-additivity, zero-monotonicity, and monotonicity. The proof of their equivalence is based on the monotonic increasing average profit function for coalitions not containing the innovator, that is, f_ns/s ≤ f_ns1/s1for all 1 ≤ s ≤ n−1. This significant property has not been discovered before and allows us to report an equivalence theorem, which sharpens the previousTheorem 1.2.

Definition 2.1. Generally speaking, a cooperative gameN, vin characteristic function form is said to be super-additive, zero-monotonic, and monotonic, respectively, if its characteristic functionvsatisfiesv∅ 0 and

ivS vT≤vS∪Tfor allS, T ⊆NwithS∩T∅super-additivity.

iivS v{i}≤vS∪ {i}for alli∈Nand allS⊆N\ {i}zero-monotonicity.

iiivS≤vTfor allS, T⊆NwithS⊆Tmonotonicity.

Theorem 2.2. For then1-person information market gameN, vof the form1.1-1.2, the following four statements are equivalent:

Super-additivity⇐⇒Zero-monotonicity⇐⇒Monotonicity⇐⇒fnn

n ≤un. 2.1

Obviously, super-additivity implies zero-monotonicity and in turn, zero-monotonicity implies monotonicity (for nonnegative games). The proof of the EquivalenceTheorem 2.2 will be based on the fundamental lemma concerning the monotonicity of averaging the profit functionf_nsof the form 1.2.

Lemma 2.3. The average function given byf_ns/s_s

j1

s−1

j−1 ·p^j·1−p^s−j·u_n−sjsatisfies ifns/s≤fns1/s1for all 1≤s≤n−1,

iif_nst≥f_ns f_ntfor all 1≤s, t≤n−1 withst≤n.

Proof ofLemma 2.3. Let 1≤s≤n−1. Concerning the cases1, note thatfn1 p·unas well asf_n2 2·p·1−p·u_n−12·p²·u_nand so, the inequalityf_n2≥2·f_n1holds due to

the fact1−p·u_n−1p·un≥un. Generally speaking, the proof is based on the combinatorial relationship _s

j−1

s−1j−1

s−1j−2 for all 2≤j≤sand proceeds as follows:

fns1 s1 ^s1

j1

s j−1

·p^j·

1−p_s1−j

·u_n−s−1j p·

1−p_s

·u_n−s p^s1·un^s

j2

s−1 j−1

s−1

j−2

·p^j·

1−p_s1−j

·u_n−s−1j p·

1−p_s

·u_n−s^s

j2

s−1 j−1

·p^j·

1−p_s1−j

·u_n−s−1j

p^s1·un^s

j2

s−1 j−2

·p^j·1−p^s1−j·u_n−s−1j p·

1−ps·u_n−s^s

j2

s−1 j−1

·p^j·

1−p_s1−j

·u_n−s−1j

p^s1·un^s−1

k1

s−1 k−1

·p^k1·

1−p_s−k

·u_n−sk

^s

j1

s−1 j−1

·p^j·

1−p_s−j

· 1−p

·u_n−s−1jp·u_n−sj

≥^s

j1

s−1 j−1

·p^j·

1−p_s−j

·u_n−sj f_ns s ,

2.2

where the relevant inequality holds because the monotonic decreasing sequence uk_k∈N satisfies1−p·u_n−s−1jp·u_n−sj ≥u_n−sjfor all 1≤j ≤s. This proves parti. Concerning partii, suppose without loss of generality, 1≤s≤t≤n−1 withst≤n. By applying part itwice, we obtain

f_nst≥st·f_nt

t f_nt s·f_nt

t ≥f_nt f_ns. 2.3 Proof ofTheorem 2.2. The super-additivity condition for disjoint, nonempty coalitionsS, T ⊆ N\ {1}not containing the innovator 1reduces tofnst≥fns fnt, whose inequality holds byLemma 2.3ii. For disjoint, nonempty coalitionsS, T ⊆Nwith 1∈T, 1/∈S, it holds thatvS∪T−vT st−1·u_n−t−1·u_ns·u_nvS∪ {1}and so, the corresponding super-additivity condition reduces tovS≤vS∪ {1}or equivalently,fns≤s·unfor all 1 ≤s≤n. ByLemma 2.3i, it is necessary and sufficient thatfnn/n≤un. This proves the equivalence super-additivity⇔f_nn/n≤u_n.

The zero-monotonicity condition for coalitionsS containing the innovator is redun- dant since u_n ≥ p · u_n. Among coalitions S not containing the innovator, the zero- monotonicity condition reduces to eitherfns1 ≥ fns fn1, whose inequality holds byLemma 2.3ii, ors·un≥fns. As before, it is necessary and sufficient thatun≥fnn/n.

Finally, note that the monotonicity condition requiresvS ≤ vS∪ {1} for allS ⊆ N\ {1},S /∅, or equivalently,f_ns≤s·u_nfor all 1≤s≤n.

3. The Core of the Information Market Game

Generally speaking, marginal contributions of players are well known as upper bounds for pay-offs according to core allocations, that is,x_i≤vN−vN\ {i}for alli∈Nand allx∈ COREN, v. Throughout the paper, given a pay-offvectorx xi_i∈N∈Rⁿ¹and a coalition S ⊆N, we denotexS

i∈Sx_i, where x∅ 0. The core allocations are selected through efficiency and group rationality. The core, however, is a set-valued solution concept, which fails to satisfy the symmetry property in that users of the same type receive identical pay-offs according to core allocations. In order to determine the single-valued solution concept called nucleolus4, being some symmetrical core allocation, our main goal is to investigate the symmetrical part of the core.

Definition 3.1. i

COREN, v

x∈Rⁿ¹|xN vN, xS≥vS∀S⊆N

. 3.1

iiThe symmetrical core allocations require equal pay-offs to users, that is,

SymCOREN, v {x x_ii∈N ∈COREN, v|x₂x₃· · ·x_n x_n1}. 3.2 Lemma 3.2. iAny gameN, vwith a nonempty core, COREN, v/∅, satisfiesvN≥vN\ {i} v{i}for alli∈N.

ii In case p 1, the core of the information market game is a singleton such that COREN, v {0, un, u_n, . . . , u_n}.

iiiIn case 0≤p <1, if the information market game possesses a nonempty core, thenb^v₁ ≥0, or equivalently,n·un≥fnn.

ivIfx xi_i∈NsatisfiesxN vNas well asx_i ≤vN−vN\ {i}for alli∈N, i /1, then the core constraintsxS≥vSare redundant for all coalitionsS⊆Nwith 1∈S.

Proof . iChoosex∈COREN, vif core is nonempty. Clearly, by3.1, for alli∈N,

vN xN xN\ {i} x_i≥vN\ {i} x_i≥vN\ {i} v{i}. 3.3 iiIn casep1, then the core-constraintsv{i} ≤xi ≤vN−vN\ {i}reduce to p·un ≤ xi ≤ un and so,xi unfor all x ∈COREN, v, and alli ∈N,i /1. Consequently, by efficiency,x₁ 0. The resulting vector0, un, u_n, . . . , u_ndoes indeed satisfy all the core constraints.

iiiIn case 0≤p <1, apply partito the information market game to conclude that b^v₁ vN−vN\ {1}≥v{1} 0 and so,b^v₁ ≥0, or equivalently,n·u_n≥f_nn.

ivUnder the given circumstances, 1∈S, together with1.3, we derive the following:

xS vN−xN\S≥vN−

i∈N\S

vN−vN\ {i} vS. 3.4

Theorem 3.3. For then1-person information market gameN, vof the form1.1-1.2with 0≤p <1, the following five statements are equivalent.

iThe core is non-empty, COREN, v/∅.

iiThe symmetrical core is non-empty, SymCOREN, v/∅.

iiib^v₁ ≥0.

ivfnn/n≤un.

v{Super-additivity, Zero-monotonicity, Monotonicity}.

The implicationi ⇒ iiiis due toLemma 3.2iii. Notice the equivalencesiii ⇔ ivas well asiv ⇔ v. The implicationii ⇒ iis trivial. It remains to show the implication iv⇒ii, the proof of which will be postponed tillSection 4.

Remark 3.4. The significant conditionf_nn/n≤u_nis equivalent tog_np≤g_n1, where the functiongn:0,1 → Ris defined by

g_n p

p·ⁿ⁻¹

k0

n−1 k

·p^k·

1−p_n−1−k

·u_k1 ∀0≤p≤1. 3.5

Note thatp is treated as a variable and that the function satisfies gn1 un. It is known that any function of the form gp p^a ·1−p^b is monotonic increasing on the interval 0, a/aband monotonic decreasing on the intervala/ab,1such that its maximum is attained bypa/abat levelga/ab a^a·b^b/ab^ab. In our framework, the functiong_npis composed as the sum ofnfunctions, each of one is monotonic increasing on the subinterval0,k1/nand monotonic decreasing on the subintervalk1/n,1such that its maximum value equalsk1^k1·n−1−k^n−1−k/nⁿ. On the final intervaln− 1/n,1, all the components are monotonic decreasing, except for the very last component given byun·pⁿ. Further investigation about the graph of the functiongnpis desirable.

4. The Nucleolus of the Information Market Game

A direct consequence ofLemma 3.2ivandLemma 2.3iis the following characterization of the symmetrical part of the core.

Corollary 4.1. iA symmetrical pay-offvector of the formxα n·un−α, α, α, . . . , α∈Rⁿ¹ is a core allocation if and only ifα≤unands·α≥fnsfor all 1≤s≤n, or equivalently,

fns

s ≤α≤un, where fns s ^s

j1

s−1 j−1

·p^j·

1−p_s−j

·u_n−sj. 4.1

iiA symmetrical pay-offvector

n·un−α, α, α, . . . , α∈ SymCOREN, v iff f_nn

n ≤α≤u_n, 4.2

where f_nn

n ⁿ

j1

n−1 j−1

·p^j·

1−p_n−j

·ujp·ⁿ⁻¹

k0

n−1 k

·p^k·

1−p_n−1−k

·u_k1. 4.3

Definition 4.2. i Define the excess of coalition S ⊆ N, S /∅, at pay-off vector x in any cooperative gameN, vbye^vS, x vS−xS. Notice that all the excesses of coalitions at core allocations are nonpositive.

iiThe excess vectorθx∈R²ⁿ⁻¹at pay-offvectorxin anyn-person gameN, vhas as its coordinates the excessese^vS, x,S⊆N,S /∅, arranged in nonincreasing order.

iiiThe nucleolus 4of a cooperative game N, v is the unique pay-off vector y of which the excess vectorθysatisfies the lexicographic orderθy≤Lθxfor any pay-off vectorxsatisfying efficiency and individual rationalityi.e.,xN vNandx_i≥v{i}for alli∈N.

ivThe surpluss^v_ijxof a playeri∈Nover another playerj∈Nat pay-offvectorx in any cooperative gameN, vis given by the maximal excess among coalitions containing playeri, but not containing playerj. That is,

s^v_ijx max

e^vS, xS⊆N, i∈S, j /∈S

. 4.4

For the purpose of the determination of the nucleolus of the information market game, the next lemma reports the maximal excess levels at symmetrical pay-offvectorsxα n·un− α, α, α, . . . , α∈Rⁿ¹.

Lemma 4.3. For then1-person information market gameN, vof the form1.1-1.2, it holds that:

ie^vS, xα −n1−s·un−αfor allS⊆ Nwith 1∈S. In caseα≤u_n, then the maximal excess among nontrivial coalitions containing player 1 equalsα−u_n attained at n-person coalitions of the formN\ {i},i /1,

iie^vS, xα f_ns−s·αfor allS⊆N,S /∅, with 1/∈S. In casef_nn/n≤α, there is no general conclusion about the maximal excess among coalitions not containing player 1.

Proof . iFor allS⊆Nwith 1∈S, it holds that

e^vS, xα vS−xαS s−1·u_n−n·u_n−n·α s−1·α

−n1−s·un−α. 4.5 Under the additional assumptionα≤un, we obtain−n1−s·un−α≤ −un−α, that is, the maximum is attained forn-person coalitions of the formN\ {i},i /1,providedS /N. On the other, for allS⊆N,S /∅, with 1∈/S, it holdse^vS, xα vS−xαS f_ns−s·α.

Theorem 4.4. Suppose that the symmetrical core of then1-person information market game is nonempty, that is,u_n≥f_nn/n. Let 1≤t≤nbe a maximizer in that

fnt un

t1 ≥ fns un

s1 ∀1≤s≤n. 4.6

Letα fnt u_n/t1andxα n·un−α, α, α, . . . , α∈Rⁿ¹.

iThen the pay-offvectorxαbelongs to the symmetrical core in thatf_nn/n≤α≤u_n. iiThe nucleolus of then1-person information market game equalsxα.

Proof . Supposen·u_n≥f_nn. The following equivalences hold:

α≤u_n iff f_nt u_n

t1 ≤u_n ifff_nt≤t·u_n iff f_nt

t ≤u_n. 4.7

ByLemma 2.3i, the latter inequality holds sincef_nt/t≤f_nn/n≤u_n. So, on the one hand, α≤un. On the other, from4.6applied tosnas well as the assumptionun ≥fnn/n, it follows that:

α fnt un

t1 ≥ fnn un

n1 ≥ fnn fnn/n

n1 fnn

n . 4.8

iiFrom partiandLemma 4.3i, on the one hand, we derive the following:

s^v₁₂xα maxe^vS, xα|S⊆N,1∈S,2∈/S max−n1−s·un−α|1≤s≤n −un−αand on the other,

s^v₂₁xα maxe^vS, xα|S⊆N,2∈S,1∈/S max

fns−s·α|1≤s≤n

α−un,

4.9

where the latter equality is due to the choice ofα. The equalitys^v₁₂y s^v₂₁yfor y xα suffices to conclude that the nucleolus is given by xα. Notice that −s^v₁₂xα un −α represents the maximal bargaining range within the core by transferring money from player 1 to player 2 starting at core allocation xαwhile remaining in the core. ByLemma 3.2iv, recall the redundancy of core constraints induced by coalitions containing player 1, so no lower bound for core allocations to player 1.

If the worth of any coalition not containing player 1 is zerofor instance, the big boss games, that is,fns 0 for all 1≤ s ≤ n, thenTheorem 4.4applies witht 1,α un/2, yielding the nucleolus to simplify toun/2·n,1,1, . . . ,1. Thus, the nucleolus pay-offto the big boss equals the aggregate pay-offto all the users.

Remark 4.5. Concerning the casetn.

Recall thatb^v₁ n·un−fnnas well asb^v_i unfor alli∈N,i /1. Thus, the casetn yieldsα fnn u_n/n1 u_n−b^v₁/n1 b^v_i −b₁^v/n1for alli∈N,i /1. In other

words, in this setting, the nucleolus coincides with the center of gravity ofn1 vectors given byb^v−β·e_i,i∈N. Hereβb^v₁ ande_i is theith standard vector inRⁿ¹. Note that, for any 1≤s≤n, the underlying conditionfnnun/n1≥fnsun/s1may be rewritten as

s·fnn−n·fns

fnn−fns

≥n−s·un. 4.10 Remark 4.6. Inspired by the description of the nucleolus as given inRemark 4.5, we review a specific subclass of cooperative games with a similar conclusion concerning the nucleolus. A cooperative gameN, vis said to be 1-convex ifv∅ 0 and its corresponding gap function g^vattains its minimum at the grand coalitionN, that is, for every coalitionS⊆N,S /∅,

0≤g^vN≤g^vS, whereg^vS

i∈S

b^v_i −vS. 4.11

For 1-convex games, its nucleolus agrees with the center of gravity of the core, of which the extreme points are given byb^v−g^vN·ei,i∈N5.

Then1-person information market game satisfiesb^v_i u_nfor alli∈N,i /1, and so, its gap functiong^vis given byg^vS b₁^vn·u_n−f_nnfor allS⊆Nwith 1∈Sandg^vS s·un−fnsotherwise. Consequently, then1-person information market game of the form1.1-1.2satisfies 1-convexity if and only if any slopeΔfns fnn−f_ns/n− s, 1 ≤ s ≤ n−1, is bounded from below by the utilityu_n in thatΔfns ≥ u_n, together with Δfn0 ≤ un provided fn0 0. Observe that the latter condition, together with Lemma 2.3i, implies the validity of4.10with reference to the case t nofTheorem 4.4.

To conclude, the 1-convexity property forn1-person information market games is part of the caset nand the current procedure for the determination of the nucleolus agrees with the known approach being the center of gravity of the non-empty core.

Remark 4.7. A cooperative game N, v is said to be 2-convex 5 if v∅ 0, and its corresponding gap functiong^vsatisfies

g^vN≤g^vS ∀S⊆N withs≥2, 4.12 g^v{i}≤g^vN≤g^v{i} g^v

j

∀i, j∈N, i /j. 4.13 Recall g^vN g^v{1} b^v₁ and g^v{i} 1 −p·un for all i /1. Together with b₁^v n·u_n−f_nn, it follows that4.13reduces to1−p·u_n≤b^v₁ ≤2·1−p·u_nor equivalently,

n−22·p

·u_n≤f_nn≤

n−1p

·u_n. 4.14

Consequently, then1-person information market game satisfies 2-convexity if and only if4.14holds as well as any slopeΔfns, 2 ≤ s ≤ n−1, is bounded from below byun. Particularly,4.10holds for all 2 ≤ s ≤ n−1. Finally, it is left to the reader to derive from 4.14the relevant inequality involvings1. That is,

fnn un

n1 ≥ fn1 un

2 providedn≥3, 0≤p <1, wherefn1 p·un. 4.15

In summary, in the setting ofTheorem 4.4, the casetnapplies ton1-person information market games, which are 2-convex. Particularly, the current procedure for the determination of the nucleolus agrees with the known approach valid for 2-convex games6.

5. The Three-Person Information Market Game

The three-person information market gameN, vwithn2is given as shown inTable 1.

Note thatb^v_i u₂fori2,3, as well asb^v₁ 2·u₂−f₂2, wheref₂2 2·p·p·u₂ 1−p·u₁. Hereb₁^v≥0 is a necessary and sufficient condition for nonemptiness of the core.

The three-person information market game is 1-convex if, besidesb^v₁ ≥0, one of the following equivalences hold:

b^v₁ ≤ 1−p

·u₂⇐⇒ u₂

u₁ ≤ 2·p

2·p1 ⇐⇒p≥ A

2, whereA u₂

u₁−u₂. 5.1

Its core is described by the constraintsx₁x2x32·u2andp·u2≤x_i≤u₂fori2,3, as well as 0≤x1 ≤b₁^v. The constraintx1 ≥0 is redundant, while the constraintb₁^v ≥0 is a necessary and sufficient condition for nonemptiness of the core. We distinguish two cases concerning the core structure, depending on the location of the core constraintx₁b^v₁ with respect to the parallel linex₁ 1−p·u2. In caseb₁^v≤1−p·u2, then the core is a triangle with three vertices 0, u2, u₂,b^v₁, u₂−b₁^v, u₂, andb^v₁, u₂, u₂−b^v₁, representing the core of a 1-convex three-person game. Its nucleolus is given by the center of the core, that isb^v₁, u₂, u₂−b^v₁/3·1,1,1.

In caseb^v₁ > 1−p·u₂, then the core has five verticesu₂·0,1,1,u₂·1−p,1, p, u2·1−p, p,1,b₁^v, p·u2,2−p·u2−b^v₁, andb^v₁,2−p·u2−b^v₁, p·u2representing the core of a convex three-person gamewith respect to its imputation set.

Concerning the condition4.6, the following equivalences holdprovided 0≤p <1:

f₂2 u₂

3 ≥ f₂1 u₂

2 ⇐⇒ u₂

u₁ ≤ 4·p

4·p1 ⇐⇒ p≥ A

4, whereA u₂

u₁−u₂. 5.2

According to the mainTheorem 4.4, to conclude with, ifp ≤ A/4, thent 1,α f21 u₂/2u₂/2 p·u₂/2 and hence, the parametric representation of the nucleolus is given byu2, u₂/2, u₂/2 u2/2·−2·p, p, p.

Ifp ≥ A/4, then t 2, α f22 u2/3 u2−b^v₁/3, and hence, the parametric representation of the nucleolus is given by0, u2, u₂−1/3·−2·b^v₁, b^v₁, b^v₁.

Ifpvaries upwards from zero tillA/4, then the nucleolus starts atu2, u₂/2, u₂/2and moves with a speed scaled byu2/2. Ifpvaries downwards from 1 tillA/4, then the nucleolus starts at0, u2, u₂and moves with a speed scaled byb^v₁ 2·1−p·1p·u₂−p·u₁. Anyhow, the nucleolus moves by two different speeds from0, u2, u₂being the full core if p 1 tillu2, u2/2, u2/2, being the center of the core ifp 0 with four vertices2·u2,0,0, u2, u₂,0,u2,0, u₂, and0, u2, u₂.

Table 1

Coalition S {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

Worth vS 0 p·u2 p·u2 u2 u2 f22 2·u2

Gap g^vS b^v₁ 1−p·u2 1−p·u2 b^v₁ b^v₁ b^v₁ b^v₁

6. The Shapley Value of the Information Market Game

Theorem 6.1. The Shapley value Sh1N, vof the innovator in then1-person information market gameN, vequals the difference between one half of the aggregate pay-offand the average worth of coalitions not containing the innovator, that is,

Sh1N, v n·un

2 − 1

n1 n s0

fns ∀i∈N, i /1,

ShiN, v 1

n·vN−Sh1N, v un

2 1

n·n1·ⁿ

s0

fns.

6.1

Proof. Putfn0 0. Using its classical formula7, the Shapley value of the innovator 1 is determined as follows:

Sh₁N, v

S⊆N\{1}

s!·n−s!

n1! ·vS∪ {1}−vS

S⊆N\{1}

s!·n−s!

n1! ·vS∪ {1}−

S⊆N\{1}

s!·n−s!

n1! ·vS

S⊆N\{1}

s!·n−s!

n1! ·s·u_n−

S⊆N\{1}

s!·n−s!

n1! ·f_ns ⁿ

s0

n s

·s!·n−s!

n1! ·s·un−ⁿ

s0

n s

·s!·n−s!

n1! ·fns ⁿ

s0

s

n1 ·un−ⁿ

s0

fns

n1 n·un

2 − 1

n1 ·ⁿ

s0

fns.

6.2

Remark 6.2. The Shapley value ShN, vis a symmetric allocation, which verifies the upper core boundu_n.

Indeed, byLemma 3.2i, it holdsf_nn/n≥f_ns/sfor all 1≤s≤nand so, 1

n·n1·ⁿ

s0

fns≤ 1

n·n1·fnn n ·ⁿ

s0

s fnn 2·n ≤ u_n

2 , 6.3

where the last inequality is due to the assumptionfnn ≤ n·un. Thus, ShiN, v ≤ un for alli∈N,i /1, whereas the Shapley value for users does not necessarily meet the lower core

boundfnn/n. For instance, for the three-person information market gamewithn2 and 0≤p <1, the following equivalences hold:

Sh₂N, v≥ f₂2

2 ⇐⇒ u₂

u₁ ≥ 4·p

4·p3 ⇐⇒p≤ 3

4 ·A, 6.4

whereA u₂/u1−u₂. By the super-additivityor zero-monotonicityof the information market game, its Shapley value satisfies individual rationality, that is, ShiN, v≥v{i}for alli∈ N. To conclude, the Shapley value of the information market game is an imputation, but not necessarily a core allocationin spite of the validity of the upper core bound for users.

7. Concluding Remarks

In this paper, we study the information market games, which have been recently introduced by Galdeano et al.1. InSection 3, we study the condition for the core to be not empty. We refer the reader toSection 4 where the nucleolus is determined through a characterization of the symmetrical part of the core. Furthermore, simple proof of the Shapley value of the information market game is given inSection 5.

Acknowledgment

The first author acknowledges financial support by the National Science Foundation of China NSFCthrough Grants nos. 71171163 and 71271171.

References

1 P. L. Galdeano, J. Oviedo, and L. G. Quintas, “Shapley value in a model of information transferal,”

International Game Theory Review, vol. 12, no. 1, pp. 19–35, 2010.

2 S. Muto, M. Nakayama, J. Potters, and S. H. Tijs, “On big boss games,” Economic Studies Quarterly, vol.

39, pp. 303–321, 1988.

3 L. G. Quintas, “How to sell Private Information,” Modelling, Measurement and Control D, vol. 11, pp.

11–28, 1995.

4 D. Schmeidler, “The nucleolus of a characteristic function game,” SIAM Journal on Applied Mathematics, vol. 17, pp. 1163–1170, 1969.

5 T. S. H. Driessen, Cooperative Games, Solutions and Applications, Kluwer Academic, Dordrecht, The Netherlands, 1988.

6 T. S. H. Driessen and D. Hou, “A note on the nucleolus for 2-convex TU games,” International Journal of Game Theory, vol. 39, no. 1, pp. 185–189, 2010.

7 L. S. Shapley, “A value for n-Person games,” Annals of Mathematics Studies, vol. 28, pp. 307–317, 1953.

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