Topological Statement of the Information Aggregation Problem in Hierarchical Games

Abstract

Two problems of rational information aggregation in hierarchical games are studied. Continuous aggregation methods are considered. Rationality is assessed by two criteria, the maximum guaranteed payoff of one of the players and the dimension of the aggregate space.

APPENDIX

Proof of Lemma 1. Let $\gamma$ be an arbitrary number strictly less than ${R_\kappa }\left ( {{\Gamma _P}} \right )$. Then there exists a strategy $u_P$ such that

$$ {\gamma < \mathop {\inf }\limits _{v \in BR\left ( {{u_P}} \right )} g\left ({{u_P}\left ( {P\left ( v \right )} \right ),v} \right )}.$$

Let us fix any of such strategies and set $\lambda = \mathop {\max }\limits _{v \in V} h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right )$ if the strategy selected is regular and ${\lambda = \mathop {\sup }\limits _{v \in V} h\left ( {{u_P}\left ({P\left ( v \right )} \right ),v} \right ) - \kappa }$ otherwise.

With such a choice of $\gamma$, $u_P$, and $\lambda$, the first item in Definition 1 is satisfied. (It is sufficient to take an arbitrary $v$ from the nonempty set $BR(u_P)$.) By virtue of the condition $\gamma < \mathop {\inf \limits _{v\in BR(u_P)}} g(u_P(P(v)),v)$, for $v \in BR\left ( {{u_P}} \right )$ one has the inequality $\gamma <g\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right )$, and by virtue of the choice of $\lambda$, for $v \notin BR\left ( {{u_P}} \right )$ one has the inequality $h\left ( {{u_P}\left ( {P\left ( v \right )}\right ),v} \right ) < \lambda$. Consequently, the second item in Definition 1 is also satisfied.

Thus, $\gamma$ is the guaranteed result of player 1 in the sense of Definition 1. By virtue of the arbitrariness of $\gamma$, this implies the desired inequality ${R_\kappa }\left ( {{\Gamma _P}}\right ) \leqslant R\left ( {{\Gamma _P}} \right )$. The proof of Lemma 1 is complete. $\quad \blacksquare$

Proof of Lemma 2. The inequality ${R_\kappa }\left ( {\Gamma _P^r} \right )\leqslant R\left ( {\Gamma _P^r} \right )$ is proved in a virtually the same way as the inequality in Lemma 1. Let us prove the reverse inequality ${R_\kappa }\left ( {\Gamma _P^r} \right ) \geqslant R\left ( {\Gamma _P^r} \right )$.

Let $\gamma$ be an arbitrary guaranteed result in the sense of Definition 1. Fix a strategy $u_P$ and a number $\lambda$ whose existence is ensured by this definition. For this strategy $v$, from the first item in this definition we have $\lambda \leqslant h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v}\right ) \leqslant \mathop {\max }\limits _{v \in V} h\left ({{u_P}\left ( {P\left ( v \right )} \right ),v} \right )$; i.e., $\lambda \leqslant \mathop {\max }\limits _{v \in V} h\left ({{u_P}\left ( {P\left ( v \right )} \right ),v} \right )$. However, then for each $v \in BR\left ( {{u_P}} \right )$ one has the inequality $h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right )\geqslant \lambda$, and by virtue of the second item in Definition 1 one has the inequality ${g\left ( {{u_P}\left ( {P\left ( v \right )}\right ),v} \right ) > \gamma }$. Consequently, ${\mathop {\inf }\limits _{v \in BR\left ( {{u_P}} \right )} g\left ( {{u_P}\left ({P\left ( v \right )} \right ),v} \right ) \geqslant \gamma }$, and so much the more

$$ {R_\kappa }\left ( {\Gamma _P^r} \right ) = \mathop {\sup }\limits _{{u_P} \in U_P^r} \mathop {\inf }\limits _{v \in BR\left ( {{u_P}} \right )} g\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right ) \geqslant \gamma . $$

By virtue of $\gamma$ being arbitrary, this implies the desired inequality ${R_\kappa }\left ( {\Gamma _P^r} \right ) \geqslant R\left ( {\Gamma _P^r} \right )$.

The proof of Lemma 2 is complete. $\quad \blacksquare$

Proof of Lemma 3. It suffices to show that ${R\left ( {\Gamma _P^r}\right )\geqslant }{ R\left ( {{\Gamma _P}}\right )}$. Let us do this.

For each $w \in W$, the set ${P^{ - 1}}\left ( w \right ) = \left \{ {v \in V:P\left ( v\right ) = w} \right \}$ is compact. Therefore, the maximum $\mathop {\max }\limits _{v \in {P^{ - 1}}\left ( w \right )} h\left ( {u,v} \right )$ is attained for all $u \in U$ and $w \in W$. Denote $f\left ({u,w} \right ) = \mathop {\max }\limits _{v \in {P^{ - 1}}\left ( w\right )} h\left ( {u,v} \right )$.

The point-set mapping ${P^{ - 1}}(w)$ is closed; i.e., so is its graph

$$ \left \{ {\left ( {w,v} \right ) \in W \times V:w = P\left ( v \right )} \right \}.$$

Consequently, for each $u \in U$ the function $f\left ( {u,w} \right )$ is upper semicontinuous on the set $W$; i.e., its subgraph

$$ \Delta \left ( u \right ) = \left \{ {\left ( {w,t} \right ) \in W \times R:t \leqslant f\left ( {u,w} \right )} \right \}$$

is closed.

However, then so is the intersection of such sets,

$$ {\Delta _0} = \bigcap \limits _{u \in U} {\Delta \left ( u \right )} = \left \{ {\left ( {w,t} \right ) \in W \times R:t \leqslant \mathop {\inf }\limits _{u \in U} f\left ( {u,w} \right )} \right \}.$$

Consequently, the minimum $\mathop {\min }\limits _{u \in U} f\left ( {u,w} \right )$ is attained at some point $u_P^p\left ( w \right )$ and, moreover, the function $\phi \left ( w \right ) = \mathop {\min }\limits _{u\in U} f\left ( {u,w} \right )$ is upper semicontinuous on the set $W$ (because $\Delta _0$ is its subgraph).

Thus, we have well defined the penalty strategy $u_P^p$, and it is regular; i.e., the maximum

$$ {\lambda _0} = \mathop {\max }\limits _{v \in V} h\left ( {u_P^p\left ( {P\left ( v \right )} \right ),v} \right ) = \mathop {\max }\limits _{w \in W} \mathop {\min }\limits _{u \in U} \mathop {\max }\limits _{v \in {P^{ - 1}}\left ( w \right )} h\left ( {u,v} \right )$$

is attained.

Now let $\gamma$ be an arbitrary guaranteed result in the game $\Gamma _P$ in the sense of Definition 1. Fix a strategy $u_P$ and a number $\lambda$ whose existence is guaranteed by this definition.

Since by construction we have

$$ {\lambda _0} = \mathop {\min }\limits _{{u_P}^{\prime} \in {U_P}} \mathop {\max }\limits _{v \in V} h\left ( {{u_P}^{\prime} \left ( {P\left ( v \right )} \right ),v} \right ) = \mathop {\max }\limits _{v \in V} h\left ( {u_P^p\left ( {P\left ( v \right )} \right ),v} \right ),$$

it follows that there exists a control $v$ for which $h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right ) \geqslant h\left ( {u_P^p\left ( {P\left ( v \right )} \right ),v} \right ) \geqslant {\lambda _0}$. Therefore, without loss of generality, we can assume that $\lambda \geqslant {\lambda _0}$. Indeed, the second item in Definition 1 is the easier to satisfy the greater the $\lambda$, while the first item is satisfied for $\lambda = {\lambda _0}$ by virtue of the inequality just established.

By virtue of the choice of $u_P$ and $\lambda$, there exists a $v$ such that $h\left ( {{u_P}\left ({P\left ( v \right )} \right ),v} \right ) \geqslant \lambda$. If the condition $h\left ( {{u_P}\left ( {P\left ( v\right )} \right ),v} \right ) \leqslant \lambda$ is satisfied for all $v$, then the strategy $u_P$ is regular—the maximum $\mathop {\max }\limits _{v\in V} h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right )$ is attained and equals $\lambda$. In this case, Lemma 3 has been proved.

Otherwise, there exists a point $v_0$ at which $h\left ( {{u_P}\left ( {P\left ({{v_0}} \right )} \right ),{v_0}} \right ) > \lambda$. Set ${w_0} = P\left ( {{v_0}} \right )$ and consider the strategy $u_P^r$ defined by the condition

$$ u_P^r\left ( w \right ) = \begin {cases} u_P\left (w_0\right )\quad &\text {if}\quad w = w_0 \\ u_P^p\left ( w \right )\quad &\text {otherwise}. \end {cases}$$

The strategy $u_P^r$ is regular. Indeed, choose a $v_1$ delivering the maximum $\mathop {\max }\limits _{v \in {P^{ -1}}\left ( {{w_0}} \right )} h\left ( {P\left ( {{w_0}} \right ),v}\right )$. Then for $v \in {P^{ -1}}\left ( {{w_0}} \right )$ we have

$$ h\left ( {u_P^r\left ( {P\left ( v \right )} \right ),v} \right ) = h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right ) \leqslant h\left ( {{u_P}\left ( {P\left ( {{v_1}} \right )} \right ),{v_1}} \right ) = h\left ( {u_P^r\left ( {P\left ( {{v_1}} \right )} \right ),{v_1}} \right ).$$

For $v \notin {P^{ - 1}}\left ( {{w_0}} \right )$, we obtain

$$ \begin {aligned} h\left ( {u_P^r\left ( {P\left ( v \right )} \right ),v} \right ) &= h\left ( {u_P^p\left ( {P\left ( v \right )} \right ),v} \right ) \leqslant {\lambda _0} \\ &< h\left ( {{u_P}\left ( {P\left ( {{v_0}} \right )} \right ),{v_0}} \right ) \leqslant h\left ( {{u_P}\left ( {P\left ( {{v_1}} \right )} \right ),{v_1}} \right ) = h\left ( {u_P^r\left ( {P\left ( {{v_1}} \right )} \right ),{v_1}} \right ). \end {aligned}$$

Thus, the least upper bound $\mathop {\sup }\limits _{v \in V}h\left ( {u_P^r\left ( {P\left ( v \right )} \right ),v} \right )$ is attained, for example, at the point $v_1$.

Let us show that both items in Definition 1 are satisfied for the strategy $u_P^r$ and the number ${\lambda _1} = h\left ( {u_P^r\left ( {P\left ({{v_1}} \right )} \right ),{v_1}} \right )$. The first item is satisfied by the definition of the number $\lambda _1$, and moreover, ${{\lambda _1} \geqslant \lambda }$ and ${\lambda _1} > {\lambda _0}$.

If $v \in {P^{ - 1}}\left ( {{w_0}} \right )$ and $h\left ( {u_P^r\left ( {P\left ( v \right )} \right ),v} \right ) = h\left ( {u_P^r\left ( {P\left ( {{v_1}} \right )}\right ),{v_1}} \right )$, then

$$ h\left ( {u_P^r\left ( {P\left ( v \right )} \right ),v} \right ) = h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right ) \geqslant \lambda$$

and, by virtue of the second item in Definition 1 (for the strategy $u_P$ and the number $\lambda$), we obtain $g\left ( {u_P^r\left ( {P\left ( v\right )}\right ),v} \right ) = g\left ( {{u_P}\left ( {P\left ( v \right )}\right ),v} \right ) > \gamma$ and the second item in Definition 1 is satisfied in this case.

If $v \in {P^{ - 1}} ( {{w_0}} )$ but $h ( {u_P^r ( {P ( v )} ),v}) \ne h ( {u_P^r ( {P ( {{v_1}} )} ),{v_1}} )$, then, by the definition of the number $\lambda _1$ and the control $v_1$, we have $h ( {u_P^r ( {P ( v )} ),v} ) < h ( {u_P^r ( {P ( {{v_1}})} ),{v_1}} ) = {\lambda _1}$, and the second item in Definition 1 is again satisfied.

Finally, if $v \notin {P^{ - 1}}\left ( {{w_0}} \right )$, then $h\left ({u_P^r\left ( {P\left ( v\right )} \right ),v} \right ) = h\left ( {u_P^p\left ( {P\left ( v\right )} \right ),v} \right ) \leqslant {\lambda _0} < \lambda$, and the second item in Definition 1 is again satisfied.

Thus, the strategy $u_P^r$ allows player 1 to securely receive the payoff $\gamma$. By virtue of $\gamma$ being arbitrary, this implies the inequality $R\left ( {\Gamma _P^r} \right ) \geqslant R\left ( {{\Gamma _P}}\right )$.

This completes the proof of Lemma 3. $\quad \blacksquare$

Proof of Theorem 1. Let us define the functions

$$ \begin {gathered} \begin {aligned} {\varphi _i}\left ( v \right ) &= \max \left \{ {g\left ( {{u_i},v} \right ) - \gamma ,0} \right \},&\quad i &= 0,1,\ldots ,k, \\ {\varphi _i}(v) &= \max \left \{ {\lambda - h\left ( {{u_i},v} \right ),0} \right \},&\quad i &= k + 1,\ldots ,n, \end {aligned} \\ \phi \left ( v \right ) = \sum \limits _{i = 1}^n {{\varphi _i}\left ( v \right ).} \end {gathered} $$

Obviously, these functions are continuous and nonnegative, and since the sets ${O_0},{O_1},\ldots ,{O_n}$ cover the space $V$, it follows that the function $\phi \left ( v \right )$ is strictly positive at each point $v$.

Choose an $n$-dimensional simplex $S$ in an $n$-dimensional Euclidean space. Let ${a_0},{a_1},\ldots ,{a_n}$ be its vertices. Consider the mapping $P:V \to S$ defined by the condition

$$ P(v) = \sum \limits _{i = 0}^n {\frac {{{\varphi _i}(v)}}{{\phi (v)}}{a_i}.} $$

The mapping $P$ is continuous.

Let $I\subset \{{0,1,\ldots ,n} \}$. If for each $j\notin I$ the condition $v \notin {O_j}$ is satisfied, then the point $P(v)$ belongs to the face of $S$ with the set of vertices $\{{{a_i},\, i \in I} \}$. Since, by condition, the multiplicity of the cover ${O_0},{O_1},\ldots ,{O_n}$ is at most $m + 1$, it follows that the image $P(v)$ of any point $v$ belongs to a face of the simplex $S$ that has at most $m + 1$ vertices, and hence the dimension does not exceed $m$.

Let $W$ be the image of the set $V$ under the mapping $P$. The set $W$ is compact and, as we have just established, embedded in an $m$-dimensional subcomplex of the simplex $S$. Consequently, the dimension of the set $W$ is at most $m$. The Euclidean space is normal and has a countable base. These properties are also inherited by its subspace $W$.

We have thus defined a certain game $\Gamma _P$. Let us construct a strategy $u_P$ in this game as follows. Let $w \in W$, and let $S_0$ be the least (by inclusion) face of $S$ containing $w$. (Of course, if the dimension of $S_0$ is greater than zero, then $w$ belongs to the interior of $S_0$.) Let $\left \{ {{a_i},\,\,i \in I} \right \}$ be the set of vertices of this face. Choose the least $i \in I$ and set ${u_P}\left ( w \right ) = {u_i}$.

Let us show that both items in Definition 1 are satisfied.

By the construction of the mapping $P$, for each $v \in {O_0}$ the least face containing the point $P\left ( v \right )$ has $a_0$ as its vertex. Consequently, by the definition of the strategy $u_P$, one has the equality ${u_P}\left ( {P\left ( v \right )} \right ) = {u_0}$, and since ${{O_0}\,} { \cap \, \Theta \ne \varnothing }$, for some point $v \in {O_0}$ we have the relation $h\left ( {{u_P}\left ( {P\left ( v \right )} \right ),v} \right ) = h\left ( {{u_0},v} \right ) \geqslant \lambda$. Thus, the first item in Definition 1 is satisfied.

Now let $v$ be an arbitrary point of the set $V$, and let $I$ be the set of all indices $i$ for which the condition $v \in {O_i}$ is satisfied. Since the sets ${O_0},{O_1},\ldots ,{O_n}$ cover $V$, it follows that the set $I$ is not empty. Then, by the construction of the mapping $P$, the set of vertices of the least face of $S$ containing the point $P\left ( v \right )$ is precisely the set $\left \{{{a_i},\,\,i \in I} \right \}$. Then, by the definition of the function $u_P$, for some $i \in I$ one has the relation ${u_P}\left ( {P\left ( v\right )} \right ) = {u_i}$. If $i \leqslant k$, then $g\left ( {{u_P}\left ( {P\left ( v \right )}\right ),v} \right ) = g\left ( {{u_i},v} \right ) > \gamma$ (because $v \in {O_i}$). Otherwise, ${h\left ( {{u_P}\left ({P\left ( v \right )} \right ),v} \right ) = h\left ( {{u_i},v} \right )< \lambda }$. Consequently, the second item in Definition 1 is satisfied as well.

Thus, by definition, $\gamma$ is the guaranteed result in the game $\Gamma _P$. The proof of Theorem 1 is complete. $\quad \blacksquare$

Proof of Theorem 4. By virtue of the inequality ${R\left ( {{\Gamma _P}} \right ) >\gamma }$, the number $\gamma$ is a guaranteed result in the game $\Gamma _P$. Fix a strategy $u_P$ and a number $\lambda$ whose existence is postulated in Definition 1. Choose a control $v_0$ for which the inequality $h\left ({{u_P}\left ( {P\left ( {{v_0}} \right )} \right ),{v_0}} \right )\geqslant \lambda$ holds (such an inequality exists by virtue of the first item in Definition 1). Set ${u_0} = {u_P}\left ( {P\left ( {{v_0}} \right )} \right )$ and

$$ {O_0}\left ( u \right ) = \left \{ {v \in V:g\left ( {{u_0},v} \right ) > \gamma } \right \}. $$

The set $O_0$ is not empty, because, by virtue of the second item in Definition 1, $g\left ( {{u_P}\left ( {P\left ( {{v_0}} \right )}\right ),{v_0}}\right )\!=\!g\left ( {{u_0},{v_0}} \right ) > \gamma$, i.e., ${v_0} \in {O_0}$. In addition, by definition, the set $O_0$ meets the set

$$ \Theta = \left \{ {v \in V:h\left ( {{u_0},v} \right ) \geqslant \lambda } \right \}. $$

For $u \in U$, define the set

$$ O^{\prime}\left ( u \right ) = \left \{ {v \in V:g\left ( {u,v} \right ) > \gamma } \right \} \cup \left \{ {v \in V:h\left ( {u,v} \right ) < \lambda } \right \}. $$

The set $O^{\prime}\left ( u \right )$ is open. Therefore, its complement $Y\left ( u \right ) =V\backslash O^{\prime}\left ( u \right )$ is closed and hence compact. Then so is the set $Y^{\prime}(u)=P(Y(u))\subset W$. Hence the set $Y^{\prime}\left ( u \right )$ is closed, and its complement $O^{\prime\prime}\left ( u\right ) = W\backslash Y^{\prime}\left ( u \right )$ is an open, possibly empty set.

However, if $u = {u_P}\left ( {P\left ( v \right )} \right )$ for some $v \in V$, then, by virtue of the second item in Definition 1, one has the inclusion $P\left ( v\right ) \in O^{\prime\prime}\left (u_P( {P\left ( v \right ))}\right )$; i.e., the corresponding set $O^{\prime\prime}\left ( u \right )$ is not empty.

As was mentioned above, we can assume that the function $P$ maps the set $V$ into the entire set $W$. In such a case, the family of open sets $O^{\prime\prime}\left ( {{u_P}\left ( {P\left ( v \right )} \right )} \right ),\,\,v \in V$ covers the compact space $W$. Hence from this family one can select a finite cover of the space $W$. Denote the sets in this cover by $O_1^{\prime \prime },\ldots ,O_n^{\prime \prime }$, and the corresponding controls by ${u_1},\ldots ,{u_n}$. Let

$$ {O_i} = \left \{ {v \in V:g\left ( {{u_i},v} \right ) > \gamma } \right \} \cup \left \{ {v \in V:h\left ( {{u_i},v} \right ) < \lambda } \right \},\quad i = 1,\ldots ,n. $$

By construction, the sets ${O_0},{O_1},\ldots ,{O_n}$ (and even the sets ${O_1},\ldots ,{O_n}$) cover $V$. Moreover, for each set $\Omega ^{\prime}\subset O_i^{\prime \prime }$, its full preimage ${P^{- 1}}\left ( {\Omega ^{\prime}} \right )$ belongs to $O_i$.

However, by assumption, the space $W$ has dimension $m$. Hence, one can inscribe a cover $\Omega _0^{\prime \prime },\Omega _1^{\prime \prime },\ldots ,\Omega _l^{\prime \prime }$ whose multiplicity is at most $m + 1$ in the cover $O_1^{\prime \prime },\ldots ,O_n^{\prime \prime }$. Then the sets ${\Omega _0} = {P^{ -1}}\left (\Omega _0^{\prime \prime }\right )$ and ${\Omega _1}=P^{-1} \left (\Omega _1^{\prime \prime }\right ),\ldots , \Omega _l=P^{-1}\left (\Omega _l^{\prime \prime }\right )$ form a cover of the set $V$ inscribed in the cover ${O_0},{O_1},\ldots ,{O_n}$ and having a multiplicity not exceeding $m + 1$.

The space $W$ is assumed to be normal and having a countable base. Therefore, for each $i$ we can define a function $\psi _i:W\to \mathbb {R}$ such that $\psi _i\left (w\right )>0$ for $w\in \Omega _i^{\prime \prime }$ and $\psi _i\left (w\right )=0$ otherwise (see [14, p. 82 of the Russian translation]). Then for the functions $\psi _i\circ P$ we have $\varphi _i\left (v\right )>0$ for $v\in O_i$ and $\varphi _i\left (v\right )=0$ for other $v$.

This completes the proof of Theorem 4. $\quad \blacksquare$