Dynamic Fisher Markets via Convex Potential

In this section, we focus on dynamic Fisher markets through the lens of a convex potential function proposed in [46]. This potential has a natural interpretation as a parameter quantifying the violation of market clearing conditions. In what follows, we use this property of the potential function to quantify the deviations induced by perturbation in market parameters like supply, budgets and buyer utilities.

The convex potential function for a static CES Fisher market is [46]

Ψ_CPF(M,p) =

n

X

j=1

w_j·p_j −X

i

b_i·lnQ_i(p), where Q_i(p) =

n

X

k=1

(a_ik)^1−c(p_k)^c

!1/c

.

Note that Qi(p) is independent of the supplies of goods and the budgets of buyers; it can be interpreted as the minimum amount of money buyer ineeds to use to earn one unit of utility [88]. Since the minimum value of ΨCPF(M,p) is not zero in general we use a normalized version Φ_CPF(M,p) := Ψ_CPF(M,p)−Ψ^∗_CPF(M), where Ψ^∗_CPF(M) :=

minpΨCPF(M,p) to apply our general framework.

We study the following tatonnement price-update rule:

p^t+1_j ← p^t_j·exp γz^t_j

, (7.1)

7.2. Dynamic Fisher Markets via Convex Potential 98 whereγ is a constant depending on market parameters.

Let Ψ^∗_CPF(M) denote the minimum value of the function Ψ_CPF(M). The following theorem, stated in a simplified format from [46], demonstrates the Price-Improvement property.

Theorem 7.2 ([46]). Let p⁰ denote the initial prices and p^∗ denote the mark et equi-librium. Suppose prices are updated according to the rule (7.1). If min_jp⁰_j/p^∗_j ≥q >0, then there exists δ=δ(q, λ)>0 such that for any timet≥0, it holds Ψ_CPF(M,p^t+1)− Ψ^∗_CPF(M) ≤ (1−δ)·(Ψ_CPF(M,p^t)−Ψ^∗_CPF(M)).

For our dynamic environment, we denote the market at time t by M^t = (u^t,b^t,w^t), and

Ψ_CPF(M^t,p^t) =

n

X

j=1

w_j^t·p^t_j − X

i

b^t_i·lnQ^t_i(p^t), where Q^t_i(p) =

n

X

k=1

(a^t_ik)^1−c(p_k)^c

!1/c

.

Let Ψ^∗,t_CPF = min_pΨ_CPF(M^t,p), and Φ_CPF(M^t,p) = Ψ_CPF(M^t,p)−Ψ^∗,t_CPF.

In the following sections we establish the Market-Perturbation property for the cases when the supplies, budgets and utility functions are dynamic.

7.2.1 Dynamic Supply

In this section, we consider the case when the supplies are changing, while buyers’

budgets and utility functions are fixed. Thus, the function Q^t_i and budget b^t_i does not change over time, and we write Q_i and b_i instead.

Proposition 7.3. A market with changing supplies, keeping other parameters fixed, satisfies the market perturbation property with ∆^t= (P +B)P

j|w_j^t+1−w^t_j|.

Proof.

The last term in the above expression can be bounded as follows: Let p^∗,t+1 denote the price vector which attains the minimum value of ΨCPF(M^t+1,p). Then

Ψ^∗,t+1_CPF =

The last inequality holds since each equilibrium price is bounded above byB, the total amount of money in the market. Thus, (Ψ^∗,t_CPF−Ψ^∗,t+1_CPF ) is bounded above by Bkε^tk. Summarizing,

Φ_CPF(M^t+1,p^t+1) ≤ (1−δ)·Φ_CPF(M^t,p^t) + (P +B)kε^tk1, i.e., ∆^t= (P+B)kε^tk¹.

7.2.2 Dynamic Budgets

In this section, we consider the case when the buyers’ budgets are changing, while supplies and buyers’ utility functions are fixed.

7.2. Dynamic Fisher Markets via Convex Potential 100 Proposition 7.4. A market with changing buyers’ budgets, keeping other parameters fixed, satisfies the market perturbation property with∆^t=C⁰P

i|b^t+1_i −b^t_i|. for a constant

Using a similar approach as in the previous section, we can bound (Ψ^∗,t_CPF−Ψ^∗,t+1_CPF ).

Ψ^∗,t+1_CPF =

Combining the above two inequalities yields

Φ_CPF(M^t+1,p^t+1) ≤ (1−δ)·Φ_CPF(M^t,p^t) + X

i

(b^t+1_i −b^t_i)·lnQ_i(p^∗,t+1) Q_i(p^t+1) . Cheung et al. [46, Section 6.3] showed that in the static market setting, if the initial prices are neither too high nor too low, then ^Q_Q^i(p∗,t+1)

i(p^t+1) has time-independent upper and lower bounds. In the dynamic market setting, we assume that there exists a constant C≥1 such that the budget of each buyerichanges within the range [b⁰_i/C, C·b⁰_i]. Let U^∗, L^∗ be the time-independent upper and lower bounds derived in [46], for the static market setting withb= (b⁰₁, . . . , b⁰_m). Following the argument in [46], their upper bound onp^t+1_k can be carried through to the dynamic market setting by increasing by a factor of C, while their lower bound onp^t+1_k can be carried through to the dynamic market setting by shrinking by a factor of 1/C; these hold similarly for the equilibrium prices. Thus,

for the dynamic market setting, we have time-independent upper and lower bounds on

Qi(p^∗,t+1)

Qi(p^t+1) of valuesC²·U^∗ and L^∗/C² respectively. Thus, by setting C⁰ := max ln(C²·U^∗)

,

ln(L^∗/C²) ,

we have

ΦCPF(M^t+1,p^t+1) ≤ (1−δ)·ΦCPF(M^t,p^t) + C⁰·X

i

b^t+1_i −b^t_i ,

i.e., ∆^t=C⁰·P

i

b^t+1_i −b^t_i .

7.2.3 Dynamic Buyer Utility

In this section, we consider the case when the buyers’ utility function are changing, while supplies and budgets are fixed. In this case, changes to utility functions induce changes to the functions Q^t_i.

Proposition 7.5. A market with changing buyers’ utility functions, keeping other pa-rameters fixed, satisfies the market perturbation property with ∆^t = 2Blnχ^t, where χ^t = maxi,j((χ^t_ij)^−1/ρ,(χ^t_ij)^1/ρ). Here χij denotes the multiplicative change in utility value a_ij.

Proof.

Φ_CPF(M^t+1,p^t+1) = Ψ_CPF(M^t+1,p^t+1)−Ψ^∗,t+1_CPF

=





n

X

j=1

wj·p^t+1_j − X

i

bi·lnQ^t_i(p^t+1) − Ψ^∗,t_CPF



 − X

i

bi·lnQ^t+1_i (p^t+1) Q^t_i(p^t+1) + (Ψ^∗,t_CPF−Ψ^∗,t+1_CPF )

= h

Ψ_CPF(M^t,p^t+1)−Ψ^∗,t_CPF

i − X

i

bi·lnQ^t+1_i (p^t+1)

Q^t_i(p^t+1) + (Ψ^∗,t_CPF−Ψ^∗,t+1_CPF )

≤ (1−δ)·Φ_CPF(M^t,p^t) − X

i

b_i·lnQ^t+1_i (p^t+1)

Q^t_i(p^t+1) + (Ψ^∗,t_CPF−Ψ^∗,t+1_CPF ).

7.3. Connections to Bounds on Revenue Loss 102

Starting from the initial utility values, eacha_ij can in each round be changed by some multiplicative factor χ^t_ij. Let χ^t = maxi,j((χ^t_ij)^−1/ρ,(χ^t_ij)^1/ρ) and χ = maxtχ^t. Note

7.3 Connections to Bounds on Revenue Loss

Up until now, we have focused on the tatonnement price update with goal of analyzing its robustness to arbitrary changes in market parameters. In the previous sections, we showed that indeed, on account of the fact that tatonnement converges linearly to equilibrium, even in the case when market parameters are subject to perturbation, tatonnement ensures that the market stays in a state ofapproximate equilibrium, where the state of the market is measured with respect to a convex potential function. This approximation however naturally depends on the magnitude of these perturbations.

The reader may recall that in Chapter 5, we established a connection between the value of the potential function in any given round to the loss incurred by any seller in the same round. For the same tatonnement updates as considered here, we showed a bound on the loss in the revenue of any seller, albeit in static markets. One can however, as well borrow this analysis and plug-in the value of the potential of a perturbed market

Im Dokument On bandit learning and pricing in markets (Seite 117-122)