A pragmatic argument against equal weighting

Abstract

We present a minimal pragmatic restriction on the interpretation of the weights in the “Equal Weight View” (and, more generally, in the “Linear Pooling” view) regarding peer disagreement and show that the view cannot respect it. Based on this result we argue against the view. The restriction is the following one: if an agent, $\hbox {i}$, assigns an equal or higher weight to another agent, $\hbox {j}$, (i.e. if $\hbox {i}$ takes $\hbox {j}$ to be as epistemically competent as him or epistemically superior to him), he must be willing—in exchange for a positive and certain payment—to accept an offer to let a completely rational and sympathetic $\hbox {j}$ choose for him whether to accept a bet with positive expected utility. If $\hbox {i}$ assigns a lower weight to $\hbox {j}$ than to himself, he must not be willing to pay any positive price for letting $\hbox {j}$ choose for him. Respecting the constraint entails, we show, that the impact of disagreement on one’s degree of belief is not independent of what the disagreement is discovered to be (i.e. not independent of j’s degree of belief).

Appendix

Theorem 1

For any credence function of $\hbox {i}$ that assigns a non-trivial probability value to the possibility that j’s degree of belief in $\hbox {P}$ is different from i’s degree of belief in $\hbox {P}$, and for any non-trivial weight, $\hbox {w}$:

• 1. Even if $\hbox {w} > 1/2$ (i.e. even if $\hbox {i}$ takes himself to be more epistemically competent than $\hbox {j}$), there always exists a bet such that $\hbox {i}$ will be willing to pay a positive amount of utility in exchange for letting (a completely rational and sympathetic) $\hbox {j}$ choose for him whether to accept this bet.

• 2. Even if $\hbox {w} \le 1/2$ (i.e. even if $\hbox {i}$ takes himself to be less epistemically competent than $\hbox {j}$), there always exists a bet such that $\hbox {i}$ will be willing to pay a positive amount of utility in exchange for avoiding passing the choice of whether to accept the bet to $\hbox {j}$.

Proof

Assume ${0}< \hbox {c}(\hbox {P})< 1$ and let us rescale i’s utility function so that the payment $\hbox {i}$ receives by choosing to accept the bet in case $\hbox {P}$ is true is 1, and the payment $\hbox {i}$ receives by choosing to accept the bet in case $\hbox {P}$ is false is 0. Thus, i’s expected utility from accepting the bet is $\hbox {c}(\hbox {P})$. If $\hbox {i}$ chooses to reject the bet he gets a payment of $0<\hbox {a}<\hbox {c}(\hbox {P})$. Thus, $\hbox {i}$ prefers accepting the bet to rejecting it.

Now let $\hbox {Y} = ``\hbox {j}$’s degree of belief in $\hbox {P}$ is lower than or equal to $\hbox {a}$”. Since $\hbox {j}$ is rational and sympathetic, in case $\hbox {Y}$ is true (i.e. in case j’s degree of belief in $\hbox {P}$ is lower than or equal to $\hbox {a}$), $\hbox {j}$ will choose to reject the bet. Similarly, in case − Y is true (i.e. in case j’s degree of belief in $\hbox {P}$ is higher than $\hbox {a}$), $\hbox {j}$ will choose to accept the bet.

Let $\hbox {L}$ be the act of letting $\hbox {j}$ choose whether to accept the bet, and let $\lnot \hbox {L}$ be the act of rejecting the offer to let $\hbox {j}$ choose whether to accept the bet. We saw that i’s expected utility from $\lnot \hbox {L}$ is just the expected utility of the bet, which is $\hbox {c}(\hbox {P})$. We can now rewrite $\hbox {c}(\hbox {P})$ using the Theorem of Total Probability applied to the partition $\{\hbox {Y}, -\hbox {Y}\}$:

• 4. $\hbox {EU}(\lnot \hbox {L}) = \hbox {c}(\hbox {P}) = \hbox {c}(\hbox {P}{\vert }-\hbox {Y})\hbox {c}(-\hbox {Y}) + \hbox {c}( \hbox {P}{\vert }\hbox {Y})\hbox {c}(\hbox {Y})$

What is the expected utility of $\hbox {L}$?

• 5. $\hbox {EU}(\hbox {L}) = \hbox {c}(\hbox {P}{\vert }{-}\, \hbox {Y})\hbox {c}({-}\, \hbox {Y}) + \hbox {ac}(\hbox {Y})$

In words: i’s expected utility from passing the choice on to $\hbox {j}$ equals the probability $\hbox {i}$ assigns to the possibility that $\hbox {j}$ will accept the bet multiplied by the expected utility of the bet from the point of view of $\hbox {i}$ after learning that $\hbox {j}$ choses to accept the bet (i.e. $\hbox {c}(\hbox {P}{\vert }-\hbox {Y})$) plus the probability i assigns to the possibility of j rejecting the bet multiplied by the expected utility of rejecting the bet (i.e. a).

$\hbox {i}$ Will choose to pass the choice on to $\hbox {j}$ iff $\hbox {EU}(\lnot \hbox {L}) \le \hbox {EU}(\hbox {L})$. Thus, it immediately follows from 4 and 5 that if $\hbox {c}(\hbox {Y}) > 0$, $\hbox {i}$ chooses to pass the choice on to $\hbox {j}$ iff $\hbox {a} > \hbox {c}(\hbox {P}{\vert }\hbox {Y})$.

Let $\hbox {Y}$* be the set of all propositions of the form “j’s degree of belief in $\hbox {P}$ is $\hbox {x}_{\mathrm{i}}$” such that $\hbox {x}_{\mathrm{i}} \le \hbox {a}$, to which $\hbox {i}$ assigns a positive probability. In other words, $\hbox {Y}$ is the disjunction of all the propositions in $\hbox {Y}$*. We assume that $\hbox {Y}$* is countable. We suspect that a similar theorem to the one we present here can be proved for the uncountable case (this becomes clear by considering the rest of the proof). However, our characterization of the LP view is not well-defined in the uncountable infinite case (in which typically each proposition gets a 0 probability) and we believe that the theoretical cost involved in the philosophical interpretation of a characterization which is mathematically well-defined also in the uncountable case outweighs the benefits associated with such a characterization. This is so since a position according to which LP is true only in the case in which i assigns positive probabilities to uncountably many hypothesis regarding j’s degree of belief in $\hbox {P}$, seems unmotivated. If LP is true, it must be true also in the countable case (indeed, also for the finite case), it seems to us. Thus, proving the theorem for the countable case is enough for the philosophical goal we have set for ourselves.

Now, since, by definition $\hbox {Y}$ is the disjunction of all the propositions in $\hbox {Y}$* (we assume, of course, that all the propositions in $\hbox {Y}$* are disjoint, i.e. that j’s degree of belief in $\hbox {P}$ is unique):

• 6. $c\left( {P{|}Y} \right) =\frac{\mathop \sum \nolimits _{X_i \epsilon {Y}^{*}} c\left( {PX_i } \right) }{c\left( Y \right) }=\frac{\mathop \sum \nolimits _{X_i \epsilon {Y}^{*}} c\left( {P|X_i } \right) c\left( {X_i } \right) }{c\left( Y \right) }$

However, from LP, for each $\hbox {X}_{\mathrm{i}}$, $\hbox {c}(\hbox {P}{\vert }\hbox {X}_{\mathrm{i}}) = \hbox {wc}(\hbox {P}) + (1-\hbox {w})\hbox {x}_{\mathrm{i}}$. Thus:

• 7. $c\left( {P{|}Y} \right) =\frac{\mathop \sum \nolimits _{X_i \epsilon Y^{*}} c\left( {X_i } \right) \left( {wc\left( p \right) +\left( {1-w} \right) x_i } \right) }{c\left( Y \right) }=wc\left( p \right) +\left( {1-w} \right) \frac{\mathop \sum \nolimits _{X_i \epsilon Y^{*}} c\left( {X_i } \right) x_i }{c\left( Y \right) }$

Let us now define $Z=\frac{\mathop \sum \nolimits _{X_i \epsilon Y^{*}} c\left( {X_i } \right) x_i }{c\left( Y \right) }$ and we get:

• 8. $\hbox {c}(\hbox {P}{\vert }\hbox {Y}) ={wc}\left( {p} \right) +\left( {1-{w}} \right) {Z}$

Notice that $\hbox {Z}$ is a weighted average of all $\hbox {x}_{\mathrm{i}}$ in $\hbox {Y}$* and thus is lower than a (or equal to it in the case in which $\hbox {Y}$* includes only one proposition: “$\hbox {j}$ believes $\hbox {P}$ to degree $\hbox {a}$”). Similarly $\hbox {c}(\hbox {P}{\vert }\hbox {Y})$ must be strictly lower than $\hbox {c}(\hbox {P})$.

We are interested in the expression “$\hbox {a} - \hbox {c}(\hbox {P}{\vert }\hbox {Y})$”. We want to show that for some values of a such that $\hbox {c}(\hbox {P})> \hbox {a} > 0$, and $\hbox {c}(\hbox {Y}) > 0$, this expression is positive and for some values it is negative independently of w. If this is the case then, independently of the weight, there are always bets such that $\hbox {EU}(-\hbox {L}) > \hbox {EU}(\hbox {L})$ and bets such that $\hbox {EU}(\hbox {L}) > \hbox {EU}(-\hbox {L})$.

${{{\underline{a -- c(P|Y) < 0}}}}$

Case 1 if there is an $\hbox {x}$* which is the lowest value for j’s degree of belief in $\hbox {P}$ to which $\hbox {i}$ assigns a positive probability, set $\hbox {a}=\hbox {x}$* (so that $\hbox {c}(\hbox {Y}) >0$). From the LP formula we get:

• 9. $\hbox {c}(\hbox {P}{\vert }\hbox {Y}) =\hbox {wc}(\hbox {P}) + (1-\hbox {w})\hbox {a}$

and since $\hbox {a} < \hbox {c}(\hbox {P})$, it immediately follows that $\hbox {c}(\hbox {P}{\vert }\hbox {Y}) > \hbox {a}$ and so $\hbox {a}$-$\hbox {c}(\hbox {P}{\vert }\hbox {Y})$ is negative.

Case 2 If there is no $\hbox {x}$* which is the lowest value for j’s degree of belief in $\hbox {P}$ to which $\hbox {i}$ assigns a positive probability, then pick an a such that a $< w_i c\left( P \right)$, and then (from equation 8):

• 10. $\hbox {c}(\hbox {P}{\vert }\hbox {Y}) = {wc}\left( {P} \right) +\left( {1-{w}} \right) {Z} > \hbox {a}$

and trivially $\hbox {a}- \hbox {c}(\hbox {P}{\vert }\hbox {Y}) < 0$ and $\hbox {c}(\hbox {Y}) > 0$ (since there is no lowest $\hbox {x}$*).²¹

${{{\underline{a -- c(P|Y) > 0}}}}$:

Let $\hbox {V}$ be the propositions “j’s degree of belief in $\hbox {P}$ is lower than $\hbox {c}(\hbox {P})$” and let $\hbox {V}$* be the set of all propositions of the form “j’s degree of belief in $\hbox {P}$ is $\hbox {x}_{\mathrm{i}}$” such that $\hbox {x}_{\mathrm{i}}< \hbox {c}(\hbox {P})$ (i.e. $\hbox {V}$ is the disjunction of all the propositions in $\hbox {V}$*). Let us now define $\hbox {Z}* = \frac{\mathop \sum \nolimits _{X_i \epsilon V^{*}} c\left( {X_i } \right) x_i }{c\left( V \right) }$ . Notice that $\hbox {Z}$* is a weighted average of all $\hbox {x}_{\mathrm{i}}$ in $\hbox {V}$* and thus is strictly lower than $\hbox {c}(\hbox {P})$.

Now set:

$$\begin{aligned} \hbox {a} = wc\left( P \right) +\left( {1-w} \right) Z^{*} \end{aligned}$$

With such an $\hbox {a}$, it follows from equation 8 that

• 11. $\hbox {a} - \hbox {c}(\hbox {P}{\vert }\hbox {Y}) = \left( {1-\hbox {w}} \right) (\hbox {Z}* - \hbox {Z})$

However, since a $< \hbox {c}(\hbox {P})$, $(\hbox {Z}* - \hbox {Z}) \ge 0$.

Case 1 there is a $\hbox {X}_{\mathrm{i}}$ in $\hbox {V}^{*}$ such that $\hbox {x}_{\mathrm{i}} > wc\left( P \right) +\left( {1-w} \right) Z^{*}$. In such a case clearly $(\hbox {Z}* - \,\hbox {Z}) > 0$ and thus $\hbox {a} - \hbox {c}(\hbox {P}{\vert }\hbox {Y}) > 0$.

Case 2 there is no such $\hbox {X}_{\mathrm{i}}$. In such a case there exists a $\hbox {X}_{\mathrm{i}}$ in $\hbox {V}$* with the highest $\hbox {x}_{\mathrm{i}}$ of all the $\hbox {x}_{\mathrm{i}}\hbox {s}$ in all the $\hbox {X}_{\mathrm{i}}\hbox {s}$ in $\hbox {V}$* (in other words there exists a hypothesis about j’s degree of belief in $\hbox {P}$ to which $\hbox {i}$ assigns a positive probability that assigns to $\hbox {j}$ the maximal degree of belief in $\hbox {P}$ which is strictly lower that $\hbox {c}(\hbox {P})$). Let us call this $\hbox {x}_{\mathrm{i}}$, $\hbox {x}^{**}$. By the definition of $\hbox {Y}$, $\hbox {c}(\hbox {P}{\vert }\hbox {Y})$ is equal for all values of a as long as a $> \hbox {x}^{**}$ and so it is always possible to find a high-enough a such that $\hbox {c}(\hbox {P})> \hbox {a} > \hbox {c}(\hbox {P}{\vert }\hbox {Y})$. In this case $\hbox {c}(\hbox {Y}) > 0$ is trivially true.

This concludes the proof. $\square$