Strategy-Proof Mechanism - Max Traffic Declaration
Definition 7.2. A mechanism is strategy-proof if truth-revelation is a dominant-strategy equilibrium
Theorem 7.1. MechanismM is strategy-proof. Equivalently, the strategy profile, (x1, ..., xn), is a dominant-strategy equilibrium.
Proof. Given any x−i, we want to provexi = xialways results in the highest utility for every playeri under all traffic cases.
Let N = {i} be the sorted player set N such that pi
xi
is in decreasing order. Let ej,−k, ∀k ∈ {0, N} and ∀j ∈ {k + 1, ..., n}, be the same as ej,−k in Definition 5.1 with{xi} replaced by
xi
. Also, let Traffick, k ∈ {0, N}, denote xj ≤ ej,−(j−1) ∀j ∈ {1, ..., k} and xj > ej,−k ∀j ∈ {k + 1, ..., n}.
Assume that playeri now plays xi = xiand has them-th priority, i.e. i = m and xi = xm. Whenm ≤ k, we have xi = xm ≤ em,−(m−1). Correspondingly,ui = xiwhich is the highest utility playeri can obtain. When m > k, we have xi = xm > em,−k andui = em,−k. If player i plays xi < em,−k, thenui = xi < em,−k = ui; if playeri plays xi ≥ em,−k, thenui = em,−k. In words, no other strategy results in higher utility. From the above, xi = xi results in the highest utility under all traffic cases and therefore is a dominant strategy of playeri.
Because the derivation above is applicable ∀i ∈ N, xi = xi is a dominant strategy of every playeri and the strategy profile, (x1, ..., xn), is a dominant-strategy equilibrium.
Having Mechanism M be at (x1, ..., xn), the spectrum allocation result is the same as previous. Thus efficiency and fairness hold. Such a dominant-strategy equilibrium is nevertheless not unique. In fact, any(x1, ..., xn) where xi ≥ xi ∀i ∈ N is also a dominant-strategy equilibrium. (This is implied in the proof of Theorem 7.1.) Given that Mechanism M is at any dominant-strategy equilibrium other than (x1, ..., xn), weighted max-min fairness is the only property attained.
Lastly, the worst performance of Mechanism M is attained when xi = ∞ ∀i ∈ N, i.e.
each player untruthfully declares that its max traffic is infinity. The corresponding utility
profile is(min(x1, e1), ..., min(xn, en)).
Theorem 7.2. In MechanismM, (min(x1, e1), ..., min(xn, en)) is the worst utility profile for all players.
Proof. We continue using the notations in Theorem 7.1. Given
x1, ..., xn
is any dominant-strategy equilibrium, we must have xi ≥ xi ∀i ∈ N. Assume that
x1, ..., xn
is such that MechanismM is under Traffick,k = 0. The allocation is
x1, ..., xk, ek+1,−k, ..., en,−k
and the utility profile is
min(x1, x1), ..., min(xk, xk), min(xk+1, ek+1,−k), ..., min(xn, en,−k) . We noticemin(xi, xi) = xi ∀i ∈ {1, ..., k} . Besides, by applying Corollary 5.1.5 to ei,−k
∀i ∈ {k + 1, ..., n}, we have pi(ei) ≥ pi(ei,−1) ≥ ... ≥ pi(ei,−k), or equivalently ei ≤ ... ≤ ei,−k. Thusmin(xi, ei) ≤ min(xi, ei,−k) ∀i ∈ {k + 1, ..., n}.
Summing up the above, we derive, ∀k = 0, (min(x1, e1), ..., min(xn, en)) ≤
min(x1, x1), ..., min(xk, xk), min(xk+1, ek+1,−k), ..., min(xn, en,−k)
. In other words, (min(x1, e1), ..., min(xn, en)), or equivalently (min(x1, e1), ..., min(xn, en)), is the worst utility profile.
Chapter 8 Conclusions
IEEE 802.22 is the first cognitive-radio-based wireless standard. IEEE 802.22 systems, operating over the licensed TV bands, utilize the spectrum sensing technique and the inter-BS coexistence mechanism to achieve an effective radio resource sharing with licensed users and other coexistent IEEE 802.22 devices as well.
We propose an efficient and fair spectrum sharing scheme for dynamic resource renting and offering (DRRO) and adaptive on demand channel contention (AODCC) in the IEEE 802.22 inter-BS coexistence mechanism. In our spectrum sharing game, all BSs always reach a Nash equilibrium where the spectrum allocation result is uniquely determined. The spectrum sharing algorithm is desirable because it achieves efficiency and fairness among all BSs. The allocation is efficient as allocative efficiency and Pareto optimality are achieved. It also meets both weighted max-min fair and weighted proportional fair criteria. By adopting this spectrum allocation result, a strategy-proof mechanism, ensuring efficiency and fairness at the truth-revealing dominant-strategy equilibrium, is designed to be applied in the more general case that max traffic requirements are private information.
To further enhance our research, there are still some aspects we need to work on. While setting the system model, we assume all information is available. Although a solution has been given for the extended case that the max traffic demands are private information, the assumption that BSs’ credit token budgets are known by each other may seem a little impractical yet. It will be our future work to design incentives for BSs to declare their budgets truthfully.
We also assume every unit of spectrum is equally important for every BS and the credit token budget is fairly allocated. Accordingly, the utility function of every BS is piecewise linear.
We will try to generalize this assumption by adopting different forms of utility functions, e.g. an exponential form or a convex form. Furthermore, a common marketplace for all BSs is considered for simplicity in our work. In order to meet the wide-area purpose of IEEE 802.22, we can design a multi-market scenario where each BS can choose to join one or more markets. This will be a very interesting extension. Finally, instead of using credit tokens, we aim to investigate a monetary-based spectrum allocation mechanism, in accordance with the proposed efficient and fair spectrum allocation, to apply in more different resource sharing schemes. With monetary transfer included in utility functions, a unique truth-revealing dominant-strategy equilibrium is possibly drawn.
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