Car Parking Multiplayer Quantum Download Fix

Developing an effective allocation policy is in the field of mechanism design that takes the game theory approach to designing economic incentives or framework, toward desired objectives in strategic settings, where players act rationally. That is why in almost all attempts to design a mechanism for allocating limited capacity, game theory is employed (for example, see Dai et al. 2005, 2006; Hall and Liu 2008; Elahi et al. 2012; Elahi 2013; Huang et al. 2013; Durango-Cohen and Li 2017; Qing et al. 2017). However, classical game theory is pushed to its limits when players are not acting rationally and/or they manipulate their decisions for achieving higher economic gains. This issue drives this study to explore the capabilities of quantum game theory for modeling the strategic reasoning of players in capacity allocation games.

Although it is more than two decades that Meyer (1999) merged game theory with quantum computing and proposed the first quantized game, the application of quantum games in supply chain management is scarce. As an example, Zhang et al. (2015) studied the service quality preference behaviors of service integrator and a service provider in service supply chain (SSC) with stochastic demand. Nash equilibrium and quantum game were used to optimize the models and it was identified that values of service quality and utilities under the quantum game equilibrium were superior compared to solutions under the Nash equilibrium, if players were preference-neutral for service qualities.

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To model an allocation game, we assume that supplier utilizes the base stock policy whereby replenishment order is placed to restore the base-stock \({\mathcal {K}}\), whenever the inventory position is below \({\mathcal {K}}\). This assumption is in congruence with many practices in the service supply chains in which a limited capacity (e.g., gates in airports, parking lots, and cranes in ports) is replenished to the base-level on a timely basis. The other general assumptions are: (i) time is discrete and is organized in a numerable and infinite succession of equally spaced instants; (ii) demands are non-negative; (iii) demand during a period is fulfilled with the on-hand stock at the beginning of that period; (iv) the available inventory is pooled to fulfill the demand of all buyers; (v) lead time is also assumed to be zero, similar to mentioned literature, for the sake of simplicity; (vi) excess inventory is carried to the next period; and (vii) without loss of generality and for the exposition simplicity, two buyers are considered in the allocation model (i.e., \(M=2\)).

The quantum structure of an allocation game is presented in the Fig. 1. The game starts from the vacuum state \(\vert vac\rangle _1 \otimes \, \vert vac\rangle _2\) which is the tensor product of single-mode vacuum states of two electromagnetic fields. First, the state passes through a unitary operator \({\widehat{J}}(\gamma )\) which is known to both buyers. Note that for an unitary operator, \({\widehat{J}} {\widehat{J}}^\dagger = 1\) where \({\widehat{J}}^\dagger \) is the adjoint operator of \({\widehat{J}}\). \(\gamma \) is a non-negative real number that is known as the squeezing parameter of the pair and can be reasonably perceived as a measure of entanglement. When \(\gamma \rightarrow \infty \) which is the infinite squeezing limit, the relevant pair will be approximate to the Einstein-Podolsky-Rosen (EPR) state. At this stage, the state of the game is \(\vert \psi _0\rangle = {\widehat{J}}(\gamma ) \vert vac\rangle _1 \, \vert vac\rangle _2\).

Accordingly, the results of \(q_1=x_1\) and \(q_2=x_2\) are obtained from the consequent final measurement which indicates that the original classical game is re-established. It can be therefore concluded that the strategic set \(S_m\) is the quantum equivalent of the classical strategic space.

Consider a scenario that a player adopts quantum strategy while the other player still uses the classical counterpart. In the context of capacity allocation, it means that the quantum buyer may inflate the order quantity based on the entanglement level hoping to gain more profit by securing bigger slice of the limited capacity (\({\mathcal {K}}=100\)). The classical player however truthfully submits order quantities which are equal to the received demands. The entanglement level (\(\gamma \)) of the quantum buyer is considered to be in range of (0, \(\frac{\pi }{3}\)). Zero entanglement reverts the problem to its classical counterpart where order quantities are just driven by demands.

We use Eq. 16 to compute order quantities of the quantum buyer which indicates that higher entanglement values result in more inflation in order quantities. From the practical perspective, quantum buyer is unlikely to order significantly higher than what is required. Therefore, assuming high values for the entanglement level undermines practicality of the proposed experiment. Nonetheless, in this study, a wider range of entanglement level is examined to portrait potential trends.

Two import insights can be gleaned by reviewing the fill rate of buyers. First, quantum buyer outperforms the classical player in achieving higher fill rates for the entire range of entanglement levels. Thus, all three policies (P, L, U) are manipulative. Second, allocation policies do not analogously react to the inflated orders. In other words, in addition to categorizing allocation policies into truth-inducing and manipulative types, it seems crucial to understand the degree that each allocation policy can be manipulated.

Reviewing the shortage cost of buyers 1 and 2 in Fig. 3\(b_1\) and \(b_2\) reveals that strategic reasoning helps the quantum buyer to secure higher fill rate and thus less shortage cost is realized. The only allocation policy that relatively protects the classical buyer from exponential growth of the shortage cost is the Uniform policy. It can be also noted that when entanglement level is greater than 0.4 for the Linear policy and 0.6 for the Proportional policy, no shortage is expected for the quantum buyer.

In Fig. 3\(d_1\) and \(d_2\), profit of buyers 1 and 2 is presented, respectively. When buyer 1 adopts quantum strategy, buyer 2 experiences a decline in profit. This decline is severe for Linear mechanism whereas Uniform is the the most robust policy against inflated orders of the quantum buyer. Under the Uniform policy, up to the entanglement level of approximately 0.6, quantum buyer has a slight increase in the profit which is proportionate to what the classical buyer is losing. Above 0.6, the impact of strategic reasoning is insignificant.

The most interesting observation in Fig. 3\(d_1\) is the pattern of profit for quantum buyer under Linear and Proportional policies. It is obvious that there is a trade-off between the boosted profit from higher value of allocated capacity and holding cost of the excess allocated capacity as a result of inflated orders. This trade-off justifies the existence of a local maximum in the profit of quantum buyer. The maximum profits of quantum buyer in the Proportional and Linear policies are 240 (\(\gamma =0.55\)) and 240 (\(\gamma =0.4\)), respectively. We can also notice that profit of quantum buyer for Uniform policy reaches to maximum of 115 in very high values of entanglement level which is less expected in the real business environment. When \(\gamma =0.5\), the quantum buyer can achieve a profit of 220 which is apparently less than both Proportional and Linear policies.

We observe that allocation policies deliver different performances when a buyer strategizing over order quantities. However, in general, both fill rate and the cumulative profit indicate that the quantum buyer outperforms the classical one in achieving higher fill rate as well as profit for the entire range of entanglement levels. This finding however may be absent for special cases where the ratio of holding cost to the selling price is extremely high which rarely occurs in the real business world. The recommendation derived from this finding should be however taken with some reservations. Note that quantum player is never placed in the position of negative profit for the Uniform policy. Whereas, both Linear and Proportional policies may result to losing money for higher entanglement levels, i.e. \(\gamma > 0.6\) and \(\gamma > 1.1\), respectively.

Any buyer may seize an opportunity of strategic reasoning for achieving higher profit when a manipulative allocation policy is used by a supplier. Therefore, it is prudent to examine a scenario where both buyers adopt quantum strategies. Settings are the same as the previous experiment but both buyers strategize over order quantities.

It is hard to imagine that a rational decision maker with reasonable cognitive abilities sticks to a single entanglement level whereas there is an opportunity to adjust the entanglement level in any period based on the past transactions. In the previous sections, we showed how each allocation policy performed given the strategic reasoning of decision makers. In this section, it is assumed that buyer 2 remains as a quantum buyer with a fixed entanglement level (\(\gamma \)) while buyer 1 as an adaptive quantum buyer adjusts the entanglement level (\(\gamma _{adaptive}\)) based on the past allocated stock.

As delineated in the Algorithm 1, for the first thirty transitions, both buyers uses the fixed entanglement level. Afterwards, in each period (t), buyer 1 computes the Coefficient of Adjusting the Entanglement level (CAE) as per the ratio of mean shortage cost and mean holding cost incurred over the previous periods (from 0 to \(t-1\)). When this ratio is greater than one, it indicates that the adaptive quantum buyer should select higher entanglement levels whereas when CAE is less than one, the buyer needs to choose smaller value for \(\gamma \).

Figure 5 illustrates the fill rate of buyers 1 and 2 for (\(a_1\)) Proportional, (\(a_2\)) Linear, and (\(a_3\)) Uniform allocation policies as well as the obtained profit (\(b_1\), \(b_2\), and \(b_3\)) for the same settings over the entanglement levels starting from zero to \(\frac{\pi }{3}\). For all allocation policies, it is obvious that the adaptive quantum buyer (buyer 1) outperforms the quantum buyer (buyer 2) from both fill rate and profit perspectives. Therefore, it is beneficial for a buyer to adopt the quantum strategies with adaptive entanglement level. In the same vein, we can observe that using the adaptive approach is more effective in smaller values of entanglement level. This is a useful insight as the likelihood of using the smaller values of entanglement level is higher. ff782bc1db