In the proposed resource allocation game, we prove that each UE(i, g) has the dominant strategy to report the true PEP,ˆe∗i,g = ei,g. The game reaches the truth-revealing dominant-strategy equilibrium ˆe∗ = e. Hence, the proposed mechanism is strategy-proof [34].
Definition 2.5. [Strategy-proofness] A mechanism is strategy-proof if the game induced by the mechanism reaches the truth-revealing dominant-strategy equilibrium.
The flow of the proofs are as follows. We first show how each UE influences the group resource demand by changing its strategy in Lemma 2.6. We also show that how the change of the group resource demand influences the resource allocation in Lemma 2.7. Based on these two lemmas, each UE is shown to have the truth-revealing dominant strategy in Theorem 2.2.
Lemma 2.6. Each UE(i, g) can influence the group resource demand Dg(ˆeg) by changing its strategyeˆi,g in the following three cases:
1. If UE (i, g)’s true resource demand is greater than the group resource demand, it cannot influence the group resource demand by increasing its strategy. It can only decrease the group resource demand by decreasing its strategy. Mathematically, if dg(ei,g) > Dg(ei,g, ˆe−i,g),
Dg(ˆeg) = Dg(ei,g, ˆe−i,g), ˆei,g > ei,g
Dg(ˆeg) ≤ Dg(ei,g, ˆe−i,g), ˆei,g < ei,g
(2.25)
2. If UE(i, g)’s true resource demand is less than the group resource demand, it can-not influence the group resource demand by decreasing its strategy. It can only
increase the group resource demand by increasing its strategy. Mathematically, if dg(ei,g) < Dg(ei,g, ˆe−i,g),
Dg(ˆeg) ≥ Dg(ei,g, ˆe−i,g), ˆei,g > ei,g
Dg(ˆeg) = Dg(ei,g, ˆe−i,g), ˆei,g < ei,g
(2.26)
3. If UE (i, g)’s true resource demand is the group resource demand, dg(ei,g) = Dg(ei,g, ˆe−i,g), it can increase/decrease the group resource demand by increasing/
decreasing its strategy.
Proof. The proofs for cases 1 and 2 are similar, so we only prove case 1. Case 3 is straightforward, so we do not show it here. In case 1, according to Lemma 2.5 and Design 2.2, the conditiondg(ei,g) > Dg(ei,g, ˆe−i,g) implies
dg(ei,g) > Dg(ei,g, ˆe−i,g) = dg(min {kg; ei,g, ˆe−i,g})
⇔ei,g > min {kg; ei,g, ˆe−i,g} (2.27)
Inequality (2.27) then implies (2.25).
Lemma 2.7. Without loss of generality, we assume that the group resource demands {Dg(ˆeg)}g∈G satisfy the weighted increasing order. We also assume that the group re-source demands{Dg(ˆeg)}g∈G are in traffic caseCj, j = 0, 1, . . . , G. The following two statements are true:
1. Forg = 1, 2, . . . , j, group g can acquire more/fewer resources by increasing/decreasing the group resource demand.
2. Forg = j + 1, j + 2, . . . , G, group g cannot acquire more resources by increasing the group resource demand. It can only acquire fewer resources by decreasing the group resource demand.
Proof. We drop the notationˆe and ˆeg for short. When the group resource demands are in traffic caseCj, groups1 to j fulfill their group resource demands but groups (j + 1) to G do not fulfill their group resource demands. According to (2.18), we know the resource allocation base resource allocation basesa1 toag−1 are not affected as well. Then according to (2.19), the resource allocation to groupg becomes
A′g = wg
Obviously, the new resource allocation basea′g in the g-th round should be greater than or equal toag, i.e.,a′g ≥ ag, since the group with the new orderg (group g if D′g/wg <
Dg+1/wg+1, or group(g+1) otherwise) must have greater group resource demand. There-fore, we have
which means groupg acquires more resources by increasing its group resource demand.
On the other hand, if group g decreases its group resource demand from Dg to Dg′ (i.e.,Dg′ < Dg), groupg will decrease its order from g to some k ≤ g. Note that groups 1 to (k − 1)’s orders are not affected, and neither are the resource allocation bases a1 to
ak−1. Then the resource allocation to groupg becomes
A′g = wg k−1
X
r=1
ar+ a′k
!
(2.31)
wherea′k is the new resource allocation base in thek-th round. Note that group k fulfills its group resource demand as in (2.28). Also note that groupg decreases its order from g to some smallerk, which implies Dg′/wg ≤ Dk/wk. We must havea′k≤ ak. Therefore,
A′g = wg k−1
X
r=1
ar+ a′k
!
≤ wg g
X
r=1
ar= Ag (2.32)
which means groupg acquires fewer resources by decreasing its group resource demand.
[Case 2:g = j + 1, j + 2, . . . , G. Group g does not fulfill its group resource demand.]
The proof for case 2 is quite similar so we do not repeat the details. The only difference is that groupg cannot acquire more resources by increasing its group resource demand.
Since group g does not fulfill its group resource demand, even if group g increases its group resource demand, the resource allocation base in theg-th round will not be affected.
Therefore, group g can only acquire fewer resources by decreasing its group resource demand.
Theorem 2.2. [Strategy-proofness] Each UE (i, g) has the truth-revealing dominant strat-egy eˆ∗i,g = ei,g that maximizes its utility. Therefore, the resource allocation game G reaches the truth-revealing dominant-strategy equilibrium ˆe∗ = e.
Proof. When UE(i, g) plays the truth-revealing strategy ˆei,g = ei,g, there are three pos-sibilities for the relationship between UE (i, g)’s true resource demand dg(ei,g) and the group resource demand Dg(ei,g, ˆe−i,g): (a1) dg(ei,g) > Dg(ei,g, ˆe−i,g), (a2) dg(ei,g) <
Dg(ei,g, ˆe−i,g), and (a3) dg(ei,g) = Dg(ei,g, ˆe−i,g). Given the group resource demand Dg(ei,g, ˆe−i,g), there are also two possibilities: (b1) Group g fulfills its group resource demand, and (b2) groupg does not fulfill its group resource demand. Therefore, there are totally six possibilities that UE(i, g)’s true resource demand dg(ei,g) is greater than/less than/equal to the group resource demand Dg(ei,g, ˆe−i,g), and group g fulfills its group
resource demand or not.
Now if UE (i, g) plays a different strategy ˆei,g 6= ei,g, it can influence the group resource demandDg(ˆe) as stated in Lemma 2.6. Also note that group g may acquire more or fewer resources when changing the group resource demands as stated in Lemma 2.7.
We take the first possibility, (a1) and (b1), for example. If UE(i, g) increases its strategy ˆ
ei,g > ei,g, the group resource demand remains unchanged (Lemma 2.6). This implies unchanged resource allocation and unchanged UE(i, g)’s utility. If UE (i, g) decreases its strategyeˆi,g < ei,g, the group resource demand decreases as well (Lemma 2.6). This implies less resource allocation (Lemma 2.7) and lower UE(i, g)’s utility (Lemma 2.4).
The other five possibilities can be analyzed in a similar way (we do not repeat here).
Therefore, we can make the following six statements that are always true:
1. If (a1) and (b1),Ag(ˆe) ≤ Ag(ei,g, ˆe−(i,g)) < dg(ei,g).
2. If (a2) and (b1),Ag(ˆe) ≥ Ag(ei,g, ˆe−(i,g)) > dg(ei,g).
3. If (a3) and (b1), Ag(ˆe) ≥ Ag(ei,g, ˆe−(i,g)) = dg(ei,g) for ˆei,g > ei,g and Ag(ˆe) ≤ Ag(ei,g, ˆe−(i,g)) = dg(ei,g) for ˆei,g < ei,g.
4. If (a1) and (b2),Ag(ˆe) ≤ Ag(ei,g, ˆe−(i,g)) < dg(ei,g).
5. If (a2) and (b2),Ag(ˆe) = Ag(ei,g, ˆe−(i,g)) > dg(ei,g).
6. If (a3) and (b2),Ag(ˆe) ≤ Ag(ei,g, ˆe−(i,g)) = dg(ei,g).
Finally, as Lemma 2.4 states that UE (i, g)’s utility function is single-peaked and maximized atdg(ei,g), we always have ui,g(ei,g, ˆe−(i,g)) ≥ ui,g(ˆe) ∀ ˆei,g 6= ei,g in all of the six possibilities. In other words, UE(i, g) has the truth-revealing dominant strategy ˆ
e∗i,g = ei,g that maximizes the utility. Therefore, game G reaches the truth-revealing dominant-strategy equilibrium ˆe∗ = e.
Remarks: There are six possibilities that the UE’s true resource demand is greater than/less than/equal to the group resource demand, and the group resource demand is satisfied or not. The UE’s utility with respect to the reported PEP in all six possibilities will be drawn
in Fig. 2.4 in Chapter 2.8.1. Recall in Chapter 2.4.4 that each UE reports the PEP in step 2, and the mechanism computes the resource allocation result in step 6. Therefore, when reporting the PEP, each UE only knows the above six possibilities but does not know in which possibility it is. Note that each UE always obtains the maximum utility in each possibility by reporting the true PEP, but may obtain less utility in some possibili-ties by reporting the false PEP. Therefore, it has no incentive to report the false PEP. The proposed mechanism is ”ex-ante” strategy-proof. Taking this one step further, after the mechanism gives the resource allocation result, each UE knows exactly in which case it is. Again, since each UE always obtains the maximum utility by reporting the true PEP, it still has no incentive to change the strategy (report false PEP). Hence, the proposed mechanism is also ”ex-post” strategy-proof. Another thing worth mentioning is that after the mechanism gives the resource allocation result, the UE knowing in which case it is may obtain the same maximum utility by reporting the false PEP. One may wonder if the UE has the incentive or any benefit to report such a false PEP? The answer is no. Since re-porting such a false PEP does not affect the resource allocation result and the UE’s utility at all, it brings no benefit and will be meaningless to play such a strategy.
Lastly, in the system meaning, the proposed mechanism enables the system to reach the equilibrium operating point where the BS collects the true PEP information from the UEs and allocates resources accordingly. The resource allocation at the truth-revealing dominant-strategy equilibrium, or shortly, the equilibrium resource allocation is given in Theorem 2.3.
Theorem 2.3. [Equilibrium resource allocation] Without loss of generality, the true group resource demands{Dg(eg)}g∈G satisfy the weighted increasing order. If the group re-source demands fall in traffic caseCj, j = 0, 1, . . . , G, the equilibrium resource alloca-tion is
A∗g(e) =
Dg(eg), g ≤ j
wg(R −Pj
k=1Dk(ek)) PG
k=j+1wk
, otherwise
(2.33)
Proof. Please refer to the proof for Design 2.4.