1 Correspondence Matrix Calculation Model

About new dynamical interpretations of entropic model of correspondence matrix calculation and Nash-Wardrop's equilibrium in Beckmann's traffic flow distribution model

1 Correspondence Matrix Calculation Model1

Assume that in some town there are districts (regions), is the number of residents living at the district , and is the number of residents working at the district . By we will denote number of residents living at the district and working at the district at the moment of time . Over course of the time numbered residents (whose quantity doesn’t alter, and is equal to ) change their apartments (homes). And we suppsose, that changes may happen only by the means of the exchange of the apartments, i.e.

 xij(t)≥0,n∑j=1xij(t)≡Li,n∑i=1xij(t)≡Wj,i,j=1,…,n. (1)

Suppose, that at the time resident lives in the district and works in the district , and the resident lives in the district and works in the district . Then - is the probability for the residents with numbers and to exchange their apartments in a period of time . It’s natural to consider that probability (in the unit time) of exchanging apartments depends only on the location of the workplaces and homes, which are exchanged. For instance, it may be considered that the “distance” between district and district is , and

 pLk,m;p,q(t)≡pLexp((ckm+cpq)\footnotesize{sum of the distances}\footnotesize{before exchange}−(cpm+ckq)\footnotesize{sum of the distances}\footnotesize{after exchange})>0.

Then, by the virtue of the ergodic theorem for discrete homogeneous Markov process with finite number of states, for all (1) we have that

 limt→∞P(xij(t)=xij,i,j=1,…,n)=
 =Z−1n,n∏i=1,j=1exp(−2cijxij)⋅(xij!)−1def=p({xij}n,ni=1,j=1),

where is the normalizing multiplier.

Here we have the case where the final distribution, which is also a stationary distribution, satisfies detailed balance condition:2

 (xkm+1)(xpq+1)⋅^p⋅pLk,m;p,q=xpmxkqp({xij}n,ni=1,j=1)pLp,m;k,q,

where

 ^p=p({x11,…,xkm+1,…,xpq+1,…,xpm−1,…,xkq−1,…,xnn}).

Distribution on a set (1) is concentrated with (see below) in a neighborhood of the most probable value , which is determined as a solution of the following entropic –linear programming problem:

 n,n∑i=1,j=1xijlnxij+2n,n∑i=1,j=1cijxij→min{xij}n,ni=1,j=1∈(???). (2)

Solution of this problem might be presented as

 xij=exp(−1−λLi−λWj−2cij),

where Lagrange multipliers (dual variables) and are determined3 from the system of equations (1). In practice we usually have some information about and . So, when we solve (2), we find

 xkm({Li,Wi}ni=1;{cij}n,ni=1,j=1)

If , , then the distribution of the probabilities on the set (1) is concentrated in neighborhood of the most probable value . More precisely:

 Missing or unrecognized delimiter for \left

Indeed, let us note, that

 ∀{xij}n,ni=1,j=1∈(???)→n,n∑i=1,j=1∂lnp({x∗ij}n,ni=1,j=1)∂xij⋅(xij−x∗ij)≤0

Thus,

 lnp({xij}n,ni=1,j=1)≤lnp({x∗ij}n,ni=1,j=1)++n,n∑i=1,j=1∂2lnp({x∗ijθ+xij⋅(1−θ)}n,ni=1,j=1)∂x2ij⋅(xij−x∗ij)22

Since

 ∂2lnp({xij}n,ni=1,j=1)∂x2ij=∂2(−n,n∑i=1,j=1xijlnxij)∂x2ij=−1xij

we have “inequality of measure concentration”:

 ∀M>0,∀{xij}n,ni=1,j=1∈(???):n,n∑i=1,j=1(xij−x∗ij)22max{xij,x∗ij}≥M
 p({xij}n,ni=1,j=1)≤e−Mp({x∗ij}n,ni=1,j=1)

2 Beckmann traffic flow distribution model4

Let us consider the oriented graph , which stands for transportation route in some town ( – nodes (vertices), – arc of the network (edges)). Let be a set of pairs inlet-outlet; – route from to , if , (it will be shown later (see example by V.I. Shvetsov) that, to specify the path it may not be enough to indicate only the set of vertices. In general, one must also specify exactly which edge, connecting the specified vertices, is chosen); – set of routes in correspondence ; – collection of all routes in the network ; – flux on the way , ; – specific costs of travel on the road , ; – flux on the arc : , where (; – specific costs of travel on the arc (generally increasing, convex, smooth functions), it is natural to assume, that . Let flows on correspondences , to be known. Then , which describes flow distribution, must lie in the set:

 X={→x≥0:∑p∈Pwxp=dw,w∈W}.

Consider a game in which each element corresponds to a considerably big ( set of players of the same type. The set of pure strategies of such player is , and profit (minus losses) is defined by the formula (a player chooses a strategy and neglects the fact, that components of the vector and hence the profit depends slightly on his choice). One can show, that Nash equilibrium is equivalent to complementarity problem, which equivalent to a solution of variation inequality, which, in its turn equivalent to a solution of convex optimization problem.

 ∀w∈W,p∈Pw→x∗p⋅(Gp(→x∗)−minq∈PwGq(→x∗))=0
 ⇕
 ∀→x∈X→⟨→G(→x∗),→x−→x∗⟩≥0
 ⇕
 Ψ(→x)=∑e∈E∑p∈Pxpδpe∫0τe(z)dz→min→x∈X. (3)

It is easy to show, that in the case being strictly monotonic transformation, the Nash-Vardrop equilibrium is unique. If are increasing functions then is unique, although, as we will see later, isn’t necessarily unique.

The route at a step player at correspondence , choose independently according the mixed strategy with probability

 Missing dimension or its units for \kern

to choose path (, and with probability to choose the same strategy as at the n-th step. Here – number of players at , who have chosen at the n-th step strategy , and can be found from the normalization condition. Multiplier describes the will to imitate and, also, the reliability of using this strategy. This multiplier notices specifics of the problem (without it there could be convergence to something different from the Nash-Wardrop equilibrium). Parameter describes “the conservatism”, while “the temperature” stands for “the risk appetite”.

Theorem 1

Let be sufficiently small, . Then , where is the minimizer from (3). Moreover, if the equilibrium is unique, then

In the experiments, conducted at the Laboratory of Experimental Economics in the Faculty of Applied Mathematics and Control, MIPT, in which students of the 5 course were involved, we observed the convergence to equilibrium and “vibrations” around it. Fluctuations should be explained, apparently, by the fact that in experiments the number of players was small and the hypothesis of a competitive market was not performed. We also observed, that for students it is more likely, than . As a result there will be convergence not to the equilibrium point, but to its neighborhood. Size of the neighborhood depends on , and the number of players.

Example 1 (Braess paradox, 1968 [15]) Let correspondence thousand cars/hour (see graphs on Figures 1 and 2). Weight of the edges is time delay (in minutes) when the flow on the edge is (thousand cars/hour). For example, in case 2 (see Figure 2): . It is natural that time delay (at each of the edge) is a growth function of flow.

The following example shows, that under the very natural conditions vector-function of cost of the travel can’t be strictly monotone:

 Missing or unrecognized delimiter for \left

This, for example, can be because of

 →G(→x)=ΘT→τ(→y),→y=Θ→x,

where describes the loading of edges (arcs) of a graph of the transport network, – vector-function of cost of the travel on the edges of transport network, – incidence matrix of edges and paths, and different vectors of flow distributions may correspond to the same vector .

Example 2 (Nonuniqueness of the equilibrium; Shvetsov, 2010).
On Figure 3 the equilibrium flow distribution is shown, for all .

Theorem 2

Let be sufficiently small, . Then and Note most of the elements of can be equal to zero.

We should notice, that Theorem 2 is a refutation (in case of the considered dynamics) of the hypothesis [16]. It states that in the case of non-unique Nash-Wardrop equilibrium, the equilibrium is more likely to realize, and it is a solution of the following linear-enthropy programming problem

 ∑w∈W∑p∈Pw(xpln(xp/xp|Pw||Pw|)−xp)→min→x∈X,Θ→x=→y∗,

where – is the unique solution of .

3 Sketch of the proof of the Theorem 1

Lemma 1

Let , where . — are some randomly chosen, but fixed parameters; . Let us consider functions

 F0(→w)=∑iαiwif(wi)∑iαiwi, and F1(→w)=∑iαif(wi)∑iαi.

Then , and the equality is attained only when

 w1=w2=…=w∗.

Proof.
The proof is based on the consequent usage of the inequality between harmonic mean and geometric mean and then Cauchy inequality.

Lemma 2

For any holds the following inequality

Proof.
Without restricting the generality, we can assume that . Then

 ⟨→G(→x(n)),E[→x(n+1)−→x(n)|→x(n)]⟩=
 =γn∑w∈Wdw⎡⎢ ⎢ ⎢ ⎢ ⎢⎣∑p∈Pwxp(n)Gp(→x(n))exp(−Gp(→x(n))T)∑p∈Pwxp(n)exp(−Gp(→x(n))T)−∑p∈PwGp(→x(n))xp(n)∑p∈Pwxp(n)⎤⎥ ⎥ ⎥ ⎥ ⎥⎦.

From Lemma 1 it follows that

 ⟨→G(→x(n)),E[→x(n+1)−→x(n)|→x(n)]⟩≤0.

And the equality can be attained only on the equilibrium vector , which can not be under considered hypothesis.

Remark 1

Lemma 2 can be more specified. At some neighborhood of the equilibrium there exists , such that

 ⟨→G(→x(n)),E[→x(n+1)−→x(n)|→x(n)]⟩≤−lγn⋅(Ψ(→x(n))−Ψ(→x∗)).

Also, with some reserves, we can change constraint to .

Lemma 3 ([25], Chapter 2.2)

Let

 ∞∑n=1γn=∞,∞∑n=1(γn)2<∞.

Then

 Ψ(→x(n))a.s.⟶n→∞Ψ(→x∗)

and if the equilibrium is unique, then also

 →x(n)a.s.⟶n→∞→x∗.

Proof.
Lemma 2 and Theorem 1 from chapter 2.2 of [25] allow us to consider only the situation, when is close to . Then by Taylor formula we have:

 E[Ψ(→x(n+1))|→x(n)]=Ψ(→x(n))+

If we take mathematical expectations from both sides of this equality, we will get that there exists sufficiently large , such that

 E(Ψ(→x(n+1)))−Ψ(→x∗)≤(1−lγn)⋅(E(Ψ(→x(n)))−Ψ(→x∗))+C⋅(γn)2.

From more general statement from [25] we get that

 E(Ψ(→x(n)))−Ψ(→x∗)⟶n→∞0,

if

 ∞∑n=1γn=∞,∞∑n=1(γn)2<∞.

From Kolmogorov inequality follows

 P(∀n≥n0→Ψ(→x(n))−Ψ(→x∗)≤ε)≥
 ≥1−ε−1⋅⎛⎝E(Ψ(→x(n0))−Ψ(→x∗))+∞∑k=n0(γk)2⎞⎠.

Which concludes the proof.
In the end we will formulate a known result, which is in high correlation with the proved one.

Theorem 3

Let5 . Then there

 P(Ψ(1NN∑n=1→x(n))−Ψmin≥Ω√N)≤2exp(−C⋅Ω),

where

Authors thank S.A. Avvakumov, I.B. Gnedkov, Y.V. Dorn, I.S. Menshikov, E.A. Nurminskiy, A.A. Shananin, V.I. Shvetsov and especially A.V. Gasnikov.

The work was supported by RFBR 10-01-00321-a, 11-01-00494-a 11-07-00162-a. The second author is partially supported by the Laboratory for Structural Methods of Data Analysis in Predictive Modeling, MIPT, RF government grant, ag. 11.G34.31.0073

Footnotes

1. This section was written, based on the works [1][11].
2. Multipliers before probabilities, for example, in the state , arise because of the number of the ways to choose the resident, living in the district and working in the district , is , and independently the number of ways to choose the resident, living in the district and working in the district , is .
3. This can be done in a different ways. For example, by Bregman’s balancing method or by Newton’s method [5]. The other way is to solve the dual problem for the entropy programming problem (2). There are a lot of different algorithms with the first order oracle (MART, GISM, etc. [4], [5]). It can be shown that most of this methods (including Bregman’s) are just barrier-multiplicative antigradient descending methods [11]. At the end (when the iteration process is achieving a sufficient small vicinity of the global minimum) it is worth to use the second order interior-point method, like Nesterov–Nemirovskii polynomial algorithm [12] (for so-called “separable” tasks).
4. This section was written, based on the works [13][24].
5. See formula (2.32), statement 2.2 and example on the page 1586 in [26]

References

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