Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set

Abstract

New axiomatic characterizations of five classes of TU-values, the classes of the weighted, positively weighted, and multiweighted Shapley values, random order values, and the Harsanyi set are presented. Between all these well-known classes exists a real subset relationship. We combine axiomatizations of all individual classes into a single theorem or corollary for all classes at once. Thereby, the axiomatizations of two neighboring classes within a theorem or corollary differ by only one axiom, which is also known as parallel axiomatization or characterization. This gives us a deeper insight into the relationships between the classes. In conjunction with marginality, a new relaxation of mutual dependence (Nowak and Radzik, Games Econ Behav 8(2):389–405,1995), called coalitional differential dependence, is the key that allows us to dispense with additivity. Additionally, we propose new axiomatizations of the above five classes, in which different versions of monotonicity, associated with strong monotonicity, are decisive. Relaxations of superweak sign symmetry (Casajus, Econ Lett 176:75–78, 2019) allow the enlargement of solution classes to go hand in hand with the weakening of the changing axiom while the other axioms remain the same for all class axiomatizations.

Appendix

1.1 Additional lemmas, used in the proofs

Lemma 4.1

(Casajus and Huettner 2008) If $i \in N$ and $v,w\in \mathbb {V}$, then $MC_i^v(S)=MC_i^w(S)$ for all $S \subseteq N\backslash \{i\}$ if and only if $\Delta _v(S \cup \{i\}) = \Delta _w(S \cup \{i\})$ for all $S \subseteq N\backslash \{i\}$.

Lemma 4.2

If $i \in N$ and $v,w\in \mathbb {V}$, then $MC_i^v(T)> MC_i^{w}(T),\,T\subseteq N\backslash \{i\}$, and $MC_i^v(S)= MC_i^{w}(S)$ for all $S \subseteq N\backslash \{i\},\, S\ne T$, if and only if $\Delta _v(S\cup \{i\})= \Delta _w(S\cup \{i\})$ and $\Delta _v(T\cup \{i\})> \Delta _w(T\cup \{i\})$.

Proof

$\Rightarrow$: Let $MC_i^v(T)> MC_i^{w}(T),\,T\subseteq N\backslash \{i\}$, and $MC_i^v(S)= MC_i^{w}(S)$ for all $S \subseteq N\backslash \{i\},\, S\ne T$. We have

$$\begin{aligned} MC_i^v(T)=\sum _{S\subseteq T}\Delta _v(S\cup \{i\})> \sum _{S\subseteq T}\Delta _w(S\cup \{i\})=MC_i^w(T). \end{aligned}$$

It follows, by Lemma 4.1, $\Delta _v(T\cup \{i\})> \Delta _w(T\cup \{i\})$.

$\Leftarrow$: Let $\Delta _v(S\cup \{i\})= \Delta _w(S\cup \{i\})$ for all $S \subseteq N\backslash \{i\},\, S\ne T$, and $\Delta _v(T\cup \{i\})> \Delta _w(T\cup \{i\}),\,T\subseteq N\backslash \{i\}$. We have, by Lemma 4.1,

$$\begin{aligned} MC_i^v(T)=\sum _{S\subseteq T}\Delta _v(S\cup \{i\})> \sum _{S\subseteq T}\Delta _w(S\cup \{i\})=MC_i^w(T). \end{aligned}$$

$\square$

Lemma 4.3

If a TU-value $\varphi$ satisfies E and MDep, then $\varphi$ also satisfies NG.

Proof

Let $\varphi$ satisfy E and MDep. We show, by contradiction, that $\varphi$ meets NG: if $\varphi$ does not satisfy NG, we have in a null game at least one $i\in N$ such that $\varphi _i(\mathbf 0 )\ne 0$. If $N=\{i\}$ we immediately have a contradiction due to E, if $|N|\ge 2$, by MDep and E, we have

$$\begin{aligned} \sum _{j \in N}\varphi _j(u_N)=\frac{\sum _{j \in N}\varphi _j(\mathbf 0 )}{\varphi _i(\mathbf 0 )}\varphi _i(u_N)\Leftrightarrow 1=0,\text { a contradiction}. \end{aligned}$$

$\square$

1.2 Proofs

1.2.1 Proof of Proposition 3.5

$\Rightarrow$: Let $\varphi$ be a TU-value that satisfies WSMon, let the preconditions be as in CSMon, and define $w := v+c\cdot u_T$. Note that $\varphi$ satisfies also Mar and thus, by Proposition 3.4, CSE. Therefore, we only have to show that $\varphi _j(w)\ge \varphi _j(v)$ for all $j\in T$.

We have $\Delta _v(S)=\Delta _w(S)$ for all $S\ne T$ and $\Delta _w(T)> \Delta _v(T)$. By Lemma 4.2, we have, $MC_j^w(T\backslash \{j\})> MC_j^{v}(T\backslash \{j\}),\,T\ni j$, and $MC_j^w(S\backslash \{j\})= MC_j^{v}(S\backslash \{j\})$ for all $S \ni j,\, S\ne T$, and the claim follows by WSMon.

$\Leftarrow$: Let $\varphi$ be a TU-value that satisfies CSMon and let the preconditions be as in WSMon. Note that $\varphi$ satisfies also CSE and thus, by Proposition 3.4, Mar. Therefore, we only have to show that $\varphi _i(w)\ge \varphi _i(v)$ for all $i\in N$ such that $MC_i^w(T)> MC_i^{v}(T)$ for a $T\subseteq N\backslash \{i\}$ and $MC_i^w(S)= MC_i^{v}(S)$ for all $S \subseteq N\backslash \{i\},\, S\ne T$.

By Lemma 4.2, we have $\Delta _w(S\cup \{i\})= \Delta _v(S\cup \{i\})$ and $\Delta _w(T\cup \{i\})> \Delta _v(T\cup \{i\})$. Thus, by $v+c\cdot u_T:= w, \,c\in \mathbb {R}_{++}$, the preconditions in CSMon are satisfied and the claim follows by CSMon. $\square$

1.2.2 Proof of Theorem 3.7

$\Rightarrow$: It is well-known that all TU-values $\varphi \in \mathcal {M}$ satisfy E and, by (4), it is obvious that NG is satisfied too. By Lemma 4.1 and (4), CSE is satisfied and therefore, by Proposition 3.4, Mar too.

We show CDDep. Let $\alpha ,\beta \in \mathbb {R},\,v,w \in \mathbb {V},\,\lambda ^{\pm } \in \Lambda ^{\pm N},\,S\subseteq N$, and $i,j\in S$ be dependent in v and w. By (2) and (4), we have for $k\in \{i,j\}$

$$\begin{aligned} \textit{M}^{\,\lambda ^{\pm }}_k(v+\alpha \cdot u_S)-\textit{M}^{\,\lambda ^{\pm }}_k(v) =\lambda ^{\pm }_{S,k}\cdot \alpha \text { and } \textit{M}^{\,\lambda ^{\pm }}_k(w+\beta \cdot u_S)-\textit{M}^{\,\lambda ^{\pm }}_k(w) =\lambda ^{\pm }_{S,k}\cdot \beta \end{aligned}$$

and CDDep is satisfied. Thus, existence is shown for the (a) part. By Remark 3.1, $\mathcal {H}$ and $\mathcal {R}$ are subsets of $\mathcal {M}$, by Dragan (1992, Theorem 4.8), the TU-values from $\mathcal {R}$ satisfy SMon, and, by Nowak and Radzik (1995, Theorem 2.4 and Remark 2.3) the TU-values from $\mathcal {S}$ and $\mathcal {PS}$ satisfy MDep. The only thing that remains to be shown in the existence part is that in the cases (b), (d), and (e) the TU values of the corresponding classes meet the related monotonicity properties:

• (b) By Lemma 4.2 and (3), CSMon is satisfied and therefore, by Proposition 3.5, WSMon too.

• (d)/(e) By Nowak and Radzik (1995, Theorem 2.1, Equation (2.2)), it is immediate that all $\varphi \in \mathcal {S}$ satisfy SMon and all $\varphi \in \mathcal {PS}$ satisfy SSMon (take $\sigma =\{N\}$).

$\Leftarrow$: (a)⁶ Let $v\in \mathbb {V}$ and $\varphi$ be a TU-value that satisfies E, NG, CDDep, and Mar and therefore, by Lemma 4.1, CSE. We show, by induction on the size $r:= |\{R\subseteq N: R \text { is essential in }v\}|$, that $\varphi \in \mathcal {M}$.

Initialization: Let $r=0$. By NG, we have $\varphi \in \mathcal {M}$.

Induction step: Assume that $\varphi \in \mathcal {M}$ holds to $\varphi$ if $r\ge 0,\,r$ arbitrary (IH). Now let precisely $r+1$ coalitions $K_\ell \subseteq N,\,1\le \ell \le r+1$, be essential in v. We define K as the intersection of all $K_\ell \subseteq N,\,1\le \ell \le r+1$:

$$\begin{aligned} K:=\bigcap _{1\le \ell \le r+1}K_\ell . \end{aligned}$$

We are dealing with two different cases: (i) $i\in N\backslash K$ and (ii) $i \in K$.

• (i) Each $i \in N\backslash K$ is contained in at most r essential coalitions $K_\ell$ in v and we have at least one essential coalition $R_i$ in v such that $i\notin R_i$. Therefore, a $v_i\in \mathbb {V}$ exists for all $i \in N\backslash K$ such that $\Delta _{v_i}(R_i)=0,\, \Delta _{v_i}(S)=\Delta _{v}(S)$ for all $S\subseteq N,\,S\ne R_i$, and $\Delta _{v}(R_i)=\Delta _{v_i}(R_i)+c_i,\,c_i\in \mathbb {R},\,c_i\ne 0$. It follows, by CSE, $\varphi _i(v)=\varphi _i(v_i)$ for all $i \in N\backslash K$ and thus, by (IH),

  $$\begin{aligned} \varphi _i(v)= \textit{M}^{\,\lambda ^{\pm }}_i(v) \text { for some }\lambda ^{\pm } \in \Lambda ^{\pm N}\text { and all }i \in N\backslash K. \end{aligned}$$

• (ii) If $K=\{i\}$, we have, by E and (i), $\varphi _i(v)= \textit{M}^{\,\lambda ^{\pm }}_i(v)$ for some $\lambda ^{\pm } \in \Lambda ^{\pm N}$. If $|K|\ge 2$, by (2), all $j\in K$ are dependent in v, in $u_K$, and in 0. We have, by NG, $\varphi _i(\mathbf 0 +u_K)-\varphi _i(\mathbf 0 )=\varphi _i(u_K)$ for all $i \in K$ and, by E, $\sum _{i \in K}\varphi _i(u_K)=1$. Thus, at least one $j\in N$ exists such that $\varphi _j(u_K)\ne 0$. Let $v'\in \mathbb {V}$ such that $\Delta _{v'}(K)=0,\, \Delta _{v'}(S)=\Delta _{v}(S)$ for all $S\subseteq N,\,S\ne K$, and therefore, by (IH), $\varphi _i(v')= \textit{M}^{\,\lambda ^{\pm }}_i(v')$ for some $\lambda ^{\pm } \in \Lambda ^{\pm N}$ and all $i\in K$. It follows for all $j\in K$ with $\varphi _j(u_K)\ne 0$:

  $$\begin{aligned}&\sum _{i \in K}\big [\varphi _i(v)-\varphi _i(v')\big ]\underset{(\mathbf CDDep )}{=}\sum _{i \in K}\bigg [\frac{\varphi _i(u_K) }{\varphi _j(u_K)} \big [\varphi _j(v)-\varphi _j(v')\big ] \bigg ] \\&\quad \underset{(\mathbf E )}{=} v(N)-v'(N)-\sum _{i \in N\backslash K}\Big [\textit{M}^{\,\lambda ^{\pm }}_i(v)-\textit{M}^{\,\lambda ^{\pm }}_i(v')\Big ] =\Delta _v(K) \\&\quad \Rightarrow \quad \varphi _j(v)= \varphi _j(u_K)\Delta _v(K)+\varphi _j(v')=\textit{M}^{\,\lambda ^{\pm }}_j(v') + c_j\Delta _v(K). \end{aligned}$$

Additionally, by CDDep, we have $\varphi _i(v)=\varphi _i(v')= \textit{M}^{\,\lambda ^{\pm }}_i(v')$ for all $i\in K$ with $\varphi _i(u_K)=0$. In total, we get $\varphi _i(v)= \textit{M}^{\,{\lambda '}^{\pm }}_i(v)$ for all $i\in K$ and a ${\lambda '}^{\pm } \in \Lambda ^{\pm N}$.

• (b) The proof is completely analogues to the proof of case (a).

• (c) $\varphi$ satisfies SMon and therefore Mar. Thus, $\varphi$ is a multiweighted Shapley value and the claim follows by Dragan (1992, Theorem 4.8).

• (d) MDep implies CDDep. Thus $\varphi$ satisfies E, CDDep, Mar, and, by Lemma 4.3, NG. Therefore we have $\varphi \in \mathcal {M}$. By Dragan (1992, Proposition 4.7), $\varphi$ satisfies Mon and hence, by Nowak and Radzik (1995, Theorem 2.4), we obtain $\varphi \in \mathcal {S}$.

• (e) By Lemma 4.3, $\varphi$ satisfies NG. Therefore, by E, NG, and SSMon, we have $\varphi _i(u_N)>0$ for all $i \in N$. Let $v\in \mathbb {V}$ and $i,j\in N$ be dependent in v. By MDep, we have

  $$\begin{aligned} \frac{\varphi _i(v)}{\varphi _i(u_N)}=\frac{\varphi _j(v)}{\varphi _j(u_N)} \end{aligned}$$

and axiom A4 ($\omega$-Mutual Dependence) in Nowak and Radzik (1995) is satisfied for the special case $\sigma =\{N\}$. By Remark 3.3 and case (a), we have $\varphi \in \mathcal {M}$. Therefore, $\varphi$ satisfies also linearity and the null player property and, by Nowak and Radzik (1995, Theorem 2.1 with $\sigma =\{N\}$), we get $\varphi \in \mathcal {PS}$. $\square$

1.2.3 Proof of Corollary 3.8

According to Theorems 3.2 and 3.7, and due to the property that $\mathcal {H}, \,\mathcal {R}$, $\mathcal {S}$, and $\mathcal {PS}$ are all subsets of $\mathcal {M}$, we only have to show uniqueness.

• (a) The result is already shown in Theorem 3.7 case (a).

• (b)/(c)/(d)/(e) Let now $\varphi ^\mathcal {H},\varphi ^\mathcal {R},\,\varphi ^\mathcal {S}$, and $\varphi ^\mathcal {PS}$ be TU-values that satisfy all mentioned properties in the related cases. If $|N|=1$, NG is implied by E. Let now $|N|\ge 2$. Obviously, Pos, Mon, and WSS each imply, together with E, NG. Also SSS together with E imply NG, which we will show by contradiction. Assume, $\varphi$ does not meet NG. By E, we have at least one $i \in N$ with $\varphi _i(\mathbf 0 )>0$. It follows, by SSS, that we have $\varphi _j\ge 0$ for all $j\in N\backslash \{i\}$ which contradicts E. Therefore, all TU-values of all classes must be multiweighted Shapley values and the claim follows by Theorem 3.2.

$\square$

1.2.4 Proof of Lemma 3.11

The first part is obvious. Let $v \in \mathbb {V}$ be a totally positive game, $S\in \Omega ^N$, and $\varphi$ be a TU-value that satisfies $\mathbf SSS ^\mathbf{Pos }$, E, N, and A. In the game $\Delta _v(S)\cdot u_S$, we have, by E, N, and $\mathbf SSS ^\mathbf{Pos }$, $\varphi _i\big (\Delta _v(S)\cdot u_S\big )\ge 0$ for all $i \in N$. By (1) and A, it follows the claim. $\square$

1.2.5 Proof of Corollary 3.13

Due to the exchangeability of the players, we have, by $\mathbf SSS ^\mathbf{NG }$, $\textit{sign} \big (\varphi _i(\mathbf 0 )\big )<0$ implies $\textit{sign} \big (\varphi _j(\mathbf 0 )\big )\le 0$. Therefore, $\mathbf SSS ^\mathbf{NG }$ and E together imply NG. Then the claim follows immediately by Lemma 3.11 and Corollary 3.8. $\square$

1.3 Logical independence

We show the logical independence of the axioms in Theorem 3.7 and Corollary 3.8.

Remark 4.4

Let $v \in \mathbb {V}$. The axioms in Theorem 3.7 are logically independent:

• E: The TU-value $\varphi :=2Sh$ satisfies all axioms but E.

• NG: Let $N:=\{1,2,\ldots ,n\},\,n\ge 2$. The TU-value $\bar{\varphi }$, defined by $\bar{\varphi }_1(v):=Sh_1(v)+1,\,\bar{\varphi }_2(v):=Sh_2(v)-1$, and $\bar{\varphi }_k(v):=Sh_k(v)$ for all $3\le k \le n$, satisfies E, Mar, WSMon, SMon, and CDDep, but not NG.

• CDDep/MDep: Let $|N|\ge 2$. The TU-value $Sh^{\xi }$, defined by (5), satisfies all axioms but CDDep and MDep.

• M/WSMon/SMon/SSMon: The equal division value$ED$, given by $ED_i:=\dfrac{v(N)}{|N|}$ for all $i \in N$, satisfies E, NG, CDDep, and MDep, but not Mar, WSMon, SMon, and SSMon.

Remark 4.5

Let $v \in \mathbb {V}$. The axioms in Corollary 3.8 are logically independent:

• E: The TU-value $\varphi :=2Sh$ satisfies all axioms but E.

• Mar: The equal division value $ED$ satisfies all axioms but Mar.

• CDDep: Let $|N|\ge 2$. The TU-value $Sh^{\xi }$, defined by (5), satisfies all axioms but CDDep.

• NG/Pos/Mon/SSS/WSS: Let $N:=\{1,2,\ldots ,n\},\,n\ge 2$. The TU-value $\bar{\varphi }$, defined by $\bar{\varphi }_1(v):=Sh_1(v)+1,\,\bar{\varphi }i_2(v):=Sh_2(v)-1$, and $\bar{\varphi }_k(v):=Sh_k(v)$ for all $3\le k \le n$, satisfies E, Mar, and CDDep, but not NG, Pos, Mon, SSS, and WSS.