War-Gaming Applications for Achieving Optimum Acquisition of Future Space Systems

This chapter describes an innovative modeling and simulation approach using newly proposed Advanced Game-based Mathematical Framework (AGMF), Unified Game-based Acquisition Framework (UGAF) and a set of War-Gaming Engines (WGEs) to address future space systems acquisition challenges. Its objective is to assist the DoD Acquisition Authority (DAA) to understand the contractor’s perspective and to seek optimum Program-and-Technical-Baseline (PTB) solution and corresponding acquisition strategy under both the perspectives of the government and the contractors. The proposed approach calls for an interdisciplinary research that involves game theory, probability and statistics, and non-linear programming. The goal of this chapter is to apply the proposed war-gaming frameworks to develop and evaluate PTB solutions and associated acquisition strategies in the context of acquisition of future space systems. Our simulation results suggest that our optimization problem for the acquisition of future space systemsmeets the affordability and innovative requirements with minimum acquisition risk.


Introduction
The U.S. Department of Defense (DoD) has recently released the Defense Innovation Initiative (DII) with the goal to "reinvigorate war-gaming" and to make DoD (best) practices more innovative [1]. In addition, DoD and U.S. Air Force have generated new acquisition regulations and initiatives to promote "Owning the Technical Baseline" (OTB) and "modular open system approach (MOSA)" as enablers for affordability [2,3]. The Aerospace Corporation has been investigating and developing war-gaming techniques to improve the DoD acquisition As discussed in Ref. [7], we envision two categories of War-Gaming Engines (WGE 2 ), namely, DAA War-Gaming Engine (DAA-WGE) and Contractor-WGE (KTR-WGE). Figure 1 depicts our vision for the two categories of war-gaming applications [7]. DAA-WGE is to be played by DAA and its stakeholders (see Figure 1(a)). KTR-WGE is to be played by potential contractors (or organizations posing as contractors), with game rules dictated by DAA and its stakeholders (see Figure 1(b)). The DAA-WGE and KTR-WGE will be developed and integrated such that DAA and its stakeholders can use them for the development and generation of optimum PTB solutions and associated acquisition strategy, respectively. Note that the optimum acquisition strategy addresses contract type, associated incentives, and RFP and source selection criteria. To achieve this goal, we define and develop the following four types of game models: • DAA-PWGE: government plays the game to select optimum PTB solutions. Past acquisition data and market survey data are used to characterize each contractor's bidding behavior. A PTB solution is selected based on minimum program execution risk and cost.
• DAA-AWGE: government plays the game to select an optimum acquisition strategy associated with a PTB solution. Past acquisition data and market survey data are used to characterize each contractor's bidding behavior. An acquisition strategy is selected based on minimum program execution risk and cost.
• KTR-PWGE: in this game, we simulate the contractors as players and the goal is to select optimum contractors' PTB solutions. Past acquisition data and market survey data are used to characterize each contractor's bidding behavior. A PTB solution is selected for each contractor based on minimum program execution risk and maximum contractor profit.
• KTR-BWGE: this game also is referred to as KTR-AWGE because the contractor's bidding strategy will be derived based on the acquisition strategy specified by the government or DAA-AWGE. The KTR-AWGE game simulates the contractors as players and the goal is to select the optimum bidding strategy associated with each contractor's selected PTB solution. Past acquisition data and market survey data are used to characterize each contractor's bidding behavior. A bidding strategy is selected based on minimum program execution risk and maximum contractor profit.
To integrate these War-Gaming Engines together, we need a unified framework that can achieve the vision shown in Figure 1. The proposed unified framework described in Ref. [7] consists of two frameworks, namely, AGMF and UGAF. The development of AGMF framework to apply war-gaming concept for "Acquisition" is a "Science." The AGMF is a framework for selecting optimum game structure and game type depending on the information available for PTB Action Space (PAS) and Acquisition Action Space (AAS). On the other hand, the development of UGAF for "Exercising" AGMF is an "Art." UGAF is used for the exercising of the AGMF to generate optimum PTB solutions and optimum acquisition strategies. The overview of these unified frameworks will be provided in the following sections.  and KTR-AWGE depending on the information available for PAS and AAS [7]. As shown in Figure 2, the framework starts the game selection by answering a question concerning the player's ability to observe other player action. As depicted in Figure 2, the DAA-PWGE and DAA-AWGE always have the static game structure since all the games will be played by the DAA and its stakeholders, with contractors as players in each game. On the other hand, the KTR-PWGE and KTR-AWGE can have either static game or dynamic game structure. The KTR-PWGE and KTR-AWGE can have a dynamic game structure when the DAA and its stakeholders assume that the one contractor can observe other contractor's action when the games are played. For dynamic game structure, the players make move based on the information released from the RFP and the players' ability (or contractor's ability) to observe other players' action through the "intelligent" information gathered on the competitors. A detailed discussion of AGMF is provided in Ref. [7].

Unified Game-based Acquisition Framework (UGAF)
The goal for UAGF is two-fold, namely, (i) play games to determine optimum PTB solution for a specified set of warfighter needs, and (ii) play games to determine the corresponding optimum acquisition strategy for a specified optimum PTB solution [7]. The optimum PTB solution  is defined as the "Architecture Solution 3 " (ARCS) for the required warfighter needs that meets the affordability and innovative requirements with minimum acquisition risk. Figure 3 describes our proposed unified framework to exercise the AGMF. It describes the processing flow for the DAA-PWGE, DAA-AWGE, KTR-PWGE and KTR-AWGE to generate optimum PTB solutions and associated optimum acquisition strategies. Figure 3 also shows seven processing boxes, in the order of execution. Detailed descriptions of Boxes #1 through #7 are described in Ref. [7].

PTB War-Gaming Engines (PWGEs)
This section provides an overview of DAA-PWGE and KTR-PWGE models. The approach presented in this section follows [8]. It focuses on static Bayesian game models with "Pure" and "Mixed" games depending on the outcomes of the market survey results. For a pure game with complete and perfect information, the contractors are "surer" of their risk assessments on the selected TEs. The risk is either "Good" or "Bad" with probability of 1 and the "Belief" and/ or "Weighting" functions for this game type are not needed. For mixed games with complete and imperfect information, the contractors are "more uncertain" of their risk assessments on TEs and the "Belief" and/or "Weighting" functions are needed for assessing TE risks. In this case, the TEs are weighted based on their priorities and using a probability density function with either uniform or triangular distribution depending on the TE's uncertainty. . UGAF: a unified approach for exercising AGMF [7]. 3 Architecture solution or ARCS is descried in terms of a set of selected "Technology Enablers" (TEs) that can provide the required system capabilities, which meet the warfighter needs. TE is a specific technology solution that meets a "capability" alone or in combination with other TEs, e.g. a telemetry communications system.
The TE's uncertainties are expressed in terms technology and market uncertainties. The definition for the system requirement types and associated PTB solution framework for classifying a PTB Solution are described in Figure 4 [8]. As an example, a Type 1 Requirement is mapped into a "Less Innovative & Conservative PTB Solution" where the "Market Uncertainty" and "Technology Uncertainty" are "Low" and "Low," respectively. Since each "Requirement Type" is associated with specific measure of technology and market uncertainties, the proposed PTB solution framework allows us to select the appropriate acquisition strategy for each "Requirement Type" and assess the technology and cost risk for each "PTB Solution Type." Figure 5 provides a PTB mapping framework to identify the "Acquisition Strategy" and risks associated with each "Requirement Type" and "PTB Solution." A detailed description of PTB System Engineering (SE) frameworks, the analytical and simulation modeling approaches for developing optimum PTB solutions can be found in Ref. [8].

Analytical and simulation modeling approach for government PTB games
This section provides an overview description of the analytical and simulation modeling approaches for PTB cooperative Bayesian games for complete information with pure and mixed strategies [8].

DAA-PWGE cooperative Bayesian games set-up for complete information with pure and mixed strategies
The DAA plays static Bayesian cooperative games with either complete and perfect information (pure game) or complete and imperfect information (mixed game) using normal-form  representation of the Bayesian games [8]. Our game models assume "N" suppliers (or contractors (KTRs)) participating in the bidding games and the availability of market survey data for which Government's "request for information" (RFI) is used to collect the required data from each contractor for assessing potential TEs identified by DAA. The contractor set is defined mathematically as The DAA defines PTB strategies involving potential architecture solutions and make them available to each supplier through RFI. The DAA estimates payoff received by each supplier for each combination of PTB strategies that could be chosen by the suppliers. The potential "I" architecture solutions set or ARCS is defined mathematically as The DAA plays complete-Bayesian game with a "Pure" or "Mixed" strategy, depending on the market survey data, to select the optimum PTB solution that can achieve "Nash" equilibrium. "Pure" game will be played if the market survey data show "complete and perfect information" for TEs. On the other hand, "Mixed" game will be played when the data show "complete and imperfect information." A "Pure" strategy, S Pure , is a strategy for a contractor "k" to map an architecture solution "i" to a PTB solution "j" defined as A "Mixed" strategy, S Mixed , is a strategy for a contractor "k" to map an architecture solution "i" to a PTB solution "j" with a "Belief" function P k i, j defined as Figure 5. PTB mapping framework: requirement type-acquisition strategy mapping [7].
The "Belief" function set or "Conditional" probability set P is defined as "the probability of selecting a PTB solution type "j" given an architecture solution "i"" The "Belief" function P k i, j must satisfy the following conditions 0 ≤ P k i, j ≤ 1 and Note that the ARCS-PTB mapping rules are based on the "Requirement Type" that is given in Figure 5. The PTB "Utility" set for a "Pure Strategy," U Pure , is defined as the Payoff and Cost Function (PCF) for selecting a pure strategy S k i, for each contractor "k," which can be expressed mathematically as: Similarly, the PTB "Utility" set for a "Mixed" strategy, U Mixed , is defined as the PCF for selecting a mixed strategy S k i, j for each contractor "k" with a "Belief" function P k i, j is defined as follow: A notional description of PCF, PCF k i, j , and PCF scoring 4 approach are provided in Ref. [8].

DAA-PWGE cooperative game with complete information and pure strategy
DAA plays the DAA-PWGE "Pure Strategy" games to "Minimize" the Cost and "Maximize" the Payoff (e.g., performance) for the selected optimum strategy, S Opt . Mathematically, DAA plays the following DAA-PTB "Pure" strategy Bayesian games 4 Ref. [9] Where S k i, j is defined as in Eq. (3) and U k i, j is given by Eq. (7). This is the "MinMax" optimization problem to search for S Opt such that, assuming that the ARCS i is the optimum solution with PTB Type 1 solution: The above optimum solution is said to achieve the Nash equilibrium, which is a stable solution to this game theoretic problem in which no individual contractor can improve their payoff by a unilateral change in his bidding behavior. The DAA-PWGE pure game algorithm is shown in Figure 6 with details provided in Ref. [8].

DAA-PWGE cooperative game with complete information and mixed strategy
Similar to the "Pure" strategy, DAA plays the DAA-PWGE "Mixed Strategy" games to "Minimize" the Cost and "Maximize" the Payoff. Mathematically, DAA plays the following DAA-PTB "Mixed" Strategy Bayesian games: Where S k i, j is defined as in Eq. (4), U k i, j is given by Eq. (8) and the "Belief" function P k i, j is given by Eq. (5). Again, this is the "MinMax" optimization problem that reaches the "Nash equilibrium" when S Opt Mixed satisfies the following condition, assuming that ARCS i is the optimum solution with PTB Type 1 solution: The DAA Plays the DAA-PTB "Mixed Strategy" games to maximize the payoff for the selected optimum strategy S Opt Mixed resulting from the optimally mapping an ARCS to a PTB Solution for a given set of "Belief Function P k i, j " defined in Eq. (5). The conditional probability that the k th supplier/contractor (KTR) selects the l th TE with a weighting factor of W l for the i th architecture solution, ARCS i , given that the ARCS i is mapped to PTB Type "j" is defined as: where The "Belief Function" set "P" for all architecture solutions, i = 1, 2,…, I, can be calculated using the following equation: Note that our team 5 has recently found that the above equation provides a better mathematical model than the one described in Ref. [8] for the belief function. Since each TE will have its own "Technology Risk" and "Market Risk", 6 Eq. (15) becomes: P k_Tech i, j and P k_Market i, j must satisfy the following conditions: 5 Our team for the FY 2017 includes the REU team funded by NSF with the following selected undergraduate students: Brittany Dyer, Claire Goldhammer, Daniel Chertock and Scott Mahan. The 2017 REU team also includes Amanda Coons, a graduate assistant, and Prof. Hien Tran, a NCSU faculty advisor. 6 Note that the terms "Technology Risk/Market Risk" and "Technology Uncertainty/Market Uncertainty" are used interchangeably in this chapter.
War-Gaming Applications for Achieving Optimum Acquisition of Future Space Systems http://dx.doi.org/10.5772/intechopen.71044 A detailed description of the calculation approach for the belief function is provided in Ref. [8]. The DAA-PWGE mixed game algorithm is shown in Figure 6 with the details provided in Ref. [8]. The calculation approach described in Ref. [8] should be modified as shown in Eq. (15). The PTB tracking tool described in Ref. [6] will be used to capture the PTB solution captured by the DAA-PWGE.

Analytical and simulation modeling approach for contractor PTB games
For KTR-PWGE model, the DAA plays game on behalf of the contractors [8]. Similar to DAA-PWGE, the KTR, actually played by DAA, plays the static Bayesian "Non-Cooperative" (NC) games with complete and imperfect information or mixed game using normal-form representation of the game. The game is NC because it is assumed that the contractors do not share their bidding information. The KTR plays game to select optimum PTB solution that can maximize profits and reduce execution risks. The KTR is assumed to play games to search for an optimum PTB solution that can achieve the "Nash" equilibrium. The KTR game is set-up as follows: • Step 1: contractor set: assume to have N contractors to play the game (see Eq. (1)).
• Step 2: contractor identifies a set of potential "I" architecture solution ARCS NC based on the requirements described in the release of RFI or RFP from a government agency: • Step 3: each architecture solution selected by a KTR will be mapped into a unique PTB solution type defined by DAA. • Step 4: the Non-Cooperative (NC) game with mixed strategy and incomplete information: The strategy for the k th contractor to map the i th architecture solution to the j th PTB solution type is performed using the following mathematical expression: • Step 5: The "Belief Function" set or conditional probability set "P" for NC games: For each contractor "k", the belief function "P Non À Coop " is defined as the probability of selecting a PTB solution type "j" given the i th architecture solution: where P k NCi, j is defined as: Similar to the DAA games, the above equation provides a better mathematical model than the one described in Ref. [8] for contractor games, and Eq. (22) must also satisfy the following condition: 0 ≤ P k NCi, j ≤ 1 and Step 6: PTB Utility Set for a "NC Mixed Strategy" is defined as U Mixed_Non_Coop . This is the PCF NC for selecting a mixed strategy S k i, j for the k th contractor. Mathematical, it is given by the following equation: Step 7: The KTR plays the following mixed game to select the optimum PTB solution: Similar to DAA-PWGE, the contractor plays KTR-PWGE to maximize his payoff or "Profit" for the selected optimum strategy S Opt KTR resulting from optimally mapping an ARCS to a PTB Solution for a given set of belief function "P Non À Coop ." The detailed KTR-PWGE mixed game algorithm is shown in Figure 6 with details provided in Ref. [8]. The PTB tracking tool described in Ref. [6] will be used to capture the PTB solution captured by the KTR-PWGE.

Acquisition-bidding War-Gaming Engines (AWGEs)
The approach for the development of Acquisition-bidding WGEs presented in this section follows Ref. [9]. An overview description of the government "Acquisition" DAA-AWGEs and contractor "Bidding" KTR-AWGE will be presented in this section for commonly used contract types, including Firmed-Fixed Price (FFP 7 ) and Cost Plus Incentive Firm (CPIF). The Fixed Price Incentive Firm (FPIF) contract type can be found in Ref. [9].

Acquisition-bidding WGE set-up
The acquisition-bidding game model assumes that there are N contractors participating in the bidding of the space system with the contractor set given by Eq. (1). The following subsections describe the game setup from the government and contractor perspectives, namely, DAA-AWGE and KTR-AWGE, respectively.

DAA-AWGE game set-up
The KTRDAA-AWGE model simulates the government's acquisition games from the government perspective based on the "Contract Type" selected based on the PTB solution obtained from DAA-PWGE models described in Section 5. The DAA-AWGE strategy is to map the optimum "Type i th PTB Solution" (PTB i ), obtained from DAA-PWGE and KTR-PWGE games described in Section 5 above, to the optimum "Acquisition Strategy" and the associated "Type i th Contract" (CT i ). Denote the government strategy as S DAA , mathematically S DAA for Bayesian games with complete and imperfect information can be written as: where p DAA i is the government "Belief Function" describing the probability that the DAA will map PTB i to CT i . It is defined as follows: Each p DAA i must satisfy the following conditions: The DAA utility function U DAA is defined as: where PCF DAA i is the government "Payoff and Cost Function" (PCF) associated with the selection of the i th strategy S DAA i . If PCF is the government estimated "contractor cost" associated with the space system being acquired (PCF DAA_KTR i ), the "Nash strategy" dictates that the optimum strategy for selecting the contract parameters is to minimize the PCF according to: If the government utility function U DAA i = 1 represents the optimum government saving strategy expressed in Eq. (30), S DAA Opt , the following condition must be true according to Nash: If PCF is the government saving associated with the space system being acquired (PCF DAA_Saving i ), "Nash strategy" dictates that the optimum contract parameters can be selected by maximizing the saving or PCF according to: If the government utility function U DAA i = 1 represents the optimum government saving strategy expressed in Eq. (32), S DAA Opt , the following condition must be true according to Nash strategy: The DAA can play non-cooperative or cooperative games depending on the "Contract Type." For example, for the FFP contract type, the DAA plays non-cooperative games if the DAA provides a clear direction on the FFP contract that the lowest bidder will be selected and there is no negotiation between the government and the selected contractor. On the contrary, the FPIF contract type requires the cooperation between DAA and the selected contractor to agree on a set of sharing ratios, and perhaps on the Point of Total Assumption (PTA) [12] as well.

KTR-AWGE game set-up
The KTR-AWGE model simulates the contractor's bidding games from the contractor perspective based on the "Contract Type" and the associated contract parameters generated from the DAA-AWGE games. Let b i be the bidding strategy for the i th contractor, the contractor strategy set for Bayesian game with complete and imperfect information, S KTR , is defined as: where p KTR i is the conditional probability that the i th contractor selects the i th bidding strategy given by: The contractor utility function U KTR is defined as: where PCF KTR i is the contractor "Payoff and Cost Function" associated with the i th contractor, KTR i , who selects the i th bidding strategy S DAA i . Since the PCF KTR i is the contractor cost associated with the space system being acquired (PCF DAA_KTR i ), "Nash strategy" dictates that the optimum bidding parameters are selected by maximizing the contractor cost function according to: If the contractor utility function U KTR i = 1 represents the optimum contractor profit strategy expressed in Eq. (38), S KTR Opt , the following condition must be true according to Nash strategy: Note that the KTR-AWGE models always assume non-cooperative games since the contractors do not share their bidding strategies among themselves.
6.2. Analytical and simulation modeling approach for government acquisition games: DAA-AWGE

DAA-AWGE for FFP contract type
The DAA-AWGE game for FFP assumes that the PTB solution obtained from the DAA-PWGE game model described in Section 5 is the "Type 1 PTB Solution" and the corresponding optimum "Contract Type" is FFP (see Figure 5). The FFP game assumes that the contractor actual costs are unknown with a cost ranges of [C min , C max ], and the actual cost has either an uniform distribution or a triangular distribution. For this game, from the government's perspective, the higher is the contractor's bid, the lower is the probability of winning the contract. For optimum acquisition strategy, the contractor needs to use the "Nash strategy" to maximize the expected profit taking into consideration both his bid and other contractors' 'expected bids. For non-optimum strategy, the contractor profit is selected by a random percentage over the target cost. The DAA strategy is to minimize contractor profits by searching for a bidding solution that will increase the number of bidders to at least two bidders for increased competition at the minimum possible price. To simplify the modeling effort, the government and the contractors are assumed to have the same risk.
For FFP, the DAA-AWGE game seeks the optimum contract parameters, including the optimum fixed price P C opt . Thus, for FFP, the i th contractor's profit function PF KTR i is defined as: where P C is the fixed price and c i is the actual production cost of the i th contractor. The government payment is the fixed price P C . The DAA-AWGE game is to minimize P C . This section provides a war-gaming model for deriving the optimum fixed price P C opt_Gov , from the government perspective. From Section 5, the optimum strategy for selecting the FFP contract parameters is defined as: From Eq. (17), the optimum government fixed price, P C opt_Gov , can be found by solving the following optimization problem: Note the actual production cost c i for the i th contractor cannot be minimized. And b(c i ) is the i th contractor bidding function given by [11] b Using calculus of variation approach, the optimum fixed price from the government perspective for uniform distribution can be shown to have the following form [9]: The expected contractor cost c i or the "Target Cost" T C can be determined from the cost analysis using the cost "S-Curve" or using the expected value of the cost distribution. For uniform case, the target cost is found to be [9]: The optimum fixed price from when the production cost c i has the triangular distributed over [C min , m, C max ] with m as the mode, can be shown to have the following form [9]: where: As point out in Ref. [9], the "Nash strategy" indicates that the optimum contractor profit is determined by the maximum expected cost C max , contractor actual production cost c i , and the number of contractors "N" participating in the bidding. Using the optimum "Nash strategy," an optimum bidder can make a smaller profit compared to the non-optimum bidders on a specific bid; however, in the long run, the optimum bidder is expected to make more profit than the non-optimum bidders since he wins more bids. The DAA-AWGE algorithm for FFP Contract Type is shown in Figure 7.

DAA-AWGE for CPIF contract type
This game assumes that the PTB solution obtained from the DAA-PWGE model described in Section 5 above is the "Type 3 PTB Solution" and the corresponding optimum "Contract Type" is CPIF (see Figure 5). The modeling development approach for the DAA-AWGE CPIF contract type is identical to FPIF approach described in Ref. [9]. The model assumes that both the government and the contractor will cooperate to maximize their minimum saving/profit. Therefore, their bargaining objective will be the maximization of the minimum outcome of the saving/profit, i.e., the "maximum" value of the saving/profit. Let PCF Gov and PCF KTR i be the final compromised saving/profit points, and PCF 0 Gov and PCF 0 KTRi be the benchmark saving/profit points for the negotiation between the government and the i th contractor, respectively. The optimum CPIF contract parameters can be obtained by solving the following optimization problem [9]: Note that PCF Gov and PCF KTR i are also defined as the government's "Cost Saving" and the i th contractor profit, respectively, and they are given by [9]: where SR G i is the government sharing ratio and A c i is the actual production cost for the i th contractor, which is unknown and as before, it is assumed to be either uniformly distributed Figure 7. DAA-AWGE modeling and simulation approach for FFP contract type [9]. over [O c , P c ], or triangularly distributed over [O c , P c ] with mode "m." Substituting PCF KTR i into PCF Gov , the government's "Cost Saving" in terms of the contract parameters can be obtained as: Substituting Eq. (52) into Eq. (49), and using the calculus of variation approach, the optimization problem can be solved by searching for the Sharing Ratios (SRs) that can maximize F i and then search for the optimum target price T P that can maximize F i . For both DAA and KTR games, we first maximize the cost function F i with respect to the contractor sharing ratio, SR C i , by solving the following equation: Note that the contractor sharing ratio ranges from 0 to 1 and the government sharing ratio for the i th contractor, SR Gi , is defined as: For DAA-AWGE game from the DAA perspective, the optimization occurs with the following partial differential equation with respect to the contractor: Note that for KTR-AWGE game from the contractor perspective, Eq. (52) becomes: Solving Eq. (50) and Eq. (52), the optimum win-win sharing ratio, SR C_Opt i , and optimum winwin target price, T P Opt i , from the DAA perspective are found as follow [9]: For CPIF, the optimum contract parameters depend on whether the contract is the under-run or over-run case. The under-run case occurs when the actual cost of the i th contractor is less than or equal to target cost, i.e., A C i ≤ T c . The over-run case occurs when A C i > T c .
• Case 1: Under-run case: For this case, the benchmark saving/profit points for the negotiation between the government and the i th contractor become: Substituting Eqs. (59) and (60) into Eqs. (57) and (57), we obtain the optimum target price, 8 T p UOpti , and the optimum win-win contractor sharing ratio, SR C UOpti , for the under-run case: Using the optimum T P UOpt i and SR C UOpt i , the optimum government payment can be calculated from: The parameter α in Eq. (59) will be selected to minimize the government payment.
• Case 2: Over-run case: For this case, the benchmark saving/profit points for the negotiation between the government and the i th contractor become: Substituting Eqs. (64) and (65) into Eqs. (57) and (58) above, we obtain the optimum target price, 9 T p OOpti , and the optimum win-win contractor sharing ratio, SR C OOpt i for the over-run case: Using the optimum T p OOpti and SR C OOpti , the optimum government payment can be calculated from the following equation: The parameter β in Eq. (65) will be selected to minimize the government payment. The optimum target price depends on the ceiling price, contractor sharing ratio, target cost and 8 Note that the optimum target price expressed in Eq. (47) indicates the optimum target price that is acceptable to Government when the optimum value of α, α Opt , is selected based on the minimum Government payment. 9 This is the optimum target price that the contractor is seeking by selecting the optimum value of β based on the maximum contractor profit.
actual cost of the i th contractor. Figure 8 describes the DAA-AWGE-CPIF Monte Carlo simulation approach to determine the optimum target price, sharing ratios, and government payment under government's perspective.
As indicated in Figure 8, the output of the DAA-AWGE CPIF model includes the average optimum values of the target fee (F T_ave ), minimum fee (F min_ave ) and maximum fee (F max_ave ), assuming there will be N optimum values for all of the selected contractors by the end of the games. The calculation of these optimum values are derived from Ref. [12] and given by the following formulas: where T P UOpt i , SR C OOpt i and SR C UOpt i are defined as above. The estimate costs O c , T c and P c are the optimistic cost estimate, target cost estimate and pessimistic cost estimate are given by the cost estimate group from engineering department or finance department or contract department depending on the organization and game rules. War-Gaming Applications for Achieving Optimum Acquisition of Future Space Systems http://dx.doi.org/10.5772/intechopen.71044 6.3. Analytical and simulation modeling approach for contractor bidding games: KTR-AWGE

KTR-AWGE for FFP contract type
The contractor bidding game, KTR-AWGE, follows the setup described in Section 6.1.2 and [9]. Similar to DAA-AWGE model for FFP, the PTB solution obtained from the KTR-PWGE game model is the "Type 1 Requirement" and the corresponding optimum "Contract Type" is FFP. For FFP, the KTR-AWGE game seeks the optimum contract parameters, including the optimum fixed price P C opt , that maximizes the contractor's profit PCF KTR i [9]: where contractor profit function, PCF KTR i , is defined as: where N, b i (c i ) and c i are defined in the above sections as the number of contractors, bidding price and associated actual production cost of the i th contractor, respectively. Using the calculus of variation approach and assuming the cost to be uniformly distributed between [C min , C max ], the solution to Eq. (72) is the optimum bidding price, b Unif_Opt i , from the contractor perspective has the same form as that from the government perspective, i.e., For the triangular distribution case, the optimum bidding price from the contractor perspective can be found in Ref. [9]. The KTR-AWGE model shows that, using the "Nash strategy," the optimum contractor bidding price is also dependent on the maximum expected cost C max , the contractor actual production cost, c i , and the number of contractors "N" participating in the bidding. The model shows that a contractor's bidding strategy is optimum when it maximizes his profit based on the maximum expected cost, the actual cost and the number of bidders. The modeling and simulation approach proposed for the FFP KTR-AWGE is to combine the above analytical models with Monte Carlo simulation. The flow chart for FFP KTR-AWGE approach is very similar to FFP DAA-AWGE and can be found in Ref. [9].

KTR-AWGE for CPIF contract type
The CPIF KTR-AWGE game described in this section follows [9]. It assumes that the PTB solution obtained from the KTR-PWGE game model is "Type 3 Requirement" and the corresponding optimum "Contract Type" is CPIF. The objective of the KTR-AWGE model is to seek the optimum "bidding" target price and the associated contractor sharing ratios for maximum contractor profit, i.e., maximum benefit from the contractor perspective. Rewrite PCF KTR i (Eq. (50)) as a function of PCF Gov as follow: The optimization problem shown in Eq. (49) becomes [9]: where PCF Gov i , PCF 0 Govi , C p , SR G i , T c , A C i and PCF 0 KTRi are as defined in Section 6.2 above.
Again, using the calculus of variation approach described in Eq. (56), the optimum "bidding" target price, T p Opt i is found to be [9]: Note that for the KTR-AWGE game, the optimum win-win sharing ratio from the contractor perspective, SR C Opt i , is identical to the DAA perspective, which is shown to have the form expressed in Eq. (57). As discussed earlier, the optimum contract parameters depend on the whether the contractor is under-run or over-run. The following paragraphs describe the approach to determine the optimum sharing ratios and the target price from the contractor perspective.
• Case 1: Under-run case: For this case, we set α = 2 and the benchmark saving/profit points for the negotiation between the government and the i th contractor become [9]: From the contractor's perspective, when α = 2 the optimum contractor sharing ratio, SR C UKOpt i , is 0% (see Eq. (62)) and the government takes 100% responsibility to pay off the profit when the contractor is under-run. The optimum bidding target price, T P KUOpt i , for this given by [9]: From the contractor perspective, the optimum price is the ceiling price. Using the optimum bidding target price T P KUOpt i and sharing ratio SR C UKOpt i , the optimum government payment can be calculated from the following Eq. [9]: • Case 2: Over-run case: For this case, we set β = 1 and the benchmark saving/profit points for the negotiation between the government and the i th contractor become [9]: The optimum bidding target price for the over-run case, T P OOpt i , is found to be [9]: Again, from the contractor perspective, the optimum price is the ceiling price. Similarly, the optimum sharing ratio, SR C_OKOpt i , for the over-run case is given by [9]: This means the contractor takes 100% responsibility when it is over-run! Using the optimum bidding target price T P KOOpt i and contractor sharing ratio SR C_OKOpt i , the government payment is given by [9]: The modeling and simulation approach for CPIF KTR-AWGE is found to be similar to FPIF KTR-AWGE with the new optimum bidding target prices, T P UOpt i and T P OOpt i , and contractor sharing ratios, SR C UOpt i and SR C OOpt i , described above. The flow chart for CPIF KTR-AWGE approach is very similar to CPIF DAA-AWGE and is shown in Figure 8.

Simulation results for sample PTB solutions and commonly used contract types
Contributors: Brittany Dyer, Claire Goldhammer, Daniel Chertock and Scott Mahan The models discussed in this chapter to evaluate PTB solutions and associated acquisition strategies for acquiring future space systems were implemented in MATLAB. The simulation results shown in this section were obtained by the 2017 REU team. In particular, we will present simulation results associated with the CPIF model for acquisition strategy. Simulation results associated with FFP contract type can be found in Ref. [13].

PTB-WGE mixed game model simulation results
The above PTB-WGE models are implemented in MATLAB. The inputs for the PTB-WGE models, which include market survey results and for risk information, potential architecture solutions describing technology combinations, and the PCF information, are manually input into the PTB-WGE program using the input dialog function in MATLAB. The belief function is calculated in a Monte Carlo simulation and aggregates individual technology risk information into a risk assessment for the architecture solution as a whole. The risk and PCF assessments are then combined to form a metric, which is used to select the optimal KTR and contract type using the PTB-ARCS mapping rules shown in Figure 5. Examples of the inputs to DAA-PWGE model are shown in Figures 9-11.  War-Gaming Applications for Achieving Optimum Acquisition of Future Space Systems http://dx.doi.org/10.5772/intechopen.71044 Figure 9 shows an example of the PCF score assigned for contractor #1 with ARCS #1. If the architecture solution set consists of 6 architecture solutions (ARCS's) as shown in Figure 10, there will be 6 PCF score sheets for each contractor. This example assumes 4 contractors, hence there will be 24-PCF score sheets input to the DAA-PWGE model. Figure 11 shows an example of the market survey results for four contractors or suppliers.
Example output of the DAA-PWGE program for mixed game model, including the optimum supplier, ARCS, and associated risk, is shown in Figure 12.

DAA-AWGE CPIF model simulation results
To average out the randomness in the optimal solutions, the acquisition and bidding models were iterated several thousand times. The CPIF program outputs the average value for each optimal contract parameter, as well as average government payment, the fee adjustment formula, and average initial conditions PCF 0 G and PCF 0 Ki . Here, we present simulation results  of four contractors and three optimal bidders. Among the optimal bidders, the limits for the randomly generated control parameters α and β are adjusted to model different bidding behavior. In this simulation, we treat contractor one as the "average" bidder. The limits for contractor two are changed so that this contractor tends to select unfavorable sharing ratios from the DAA perspective; hence, contractor two is "non-cooperative." On the other hand, contractor three tends to select sharing ratios favorable to the DAA and is called the "cooperative" bidder. The precise limits for these control parameters are depicted in Figure 13. Figures 14 and 15 show the Graphical User Interface (GUI) for DAA-AWGE CPIF input parameters and program output, respectively. The simulation results depicting average government payments and savings as well as sharing rations are shown in Figures 16-18. Contract three, the "cooperative" bidder, gives the DAA more savings than the other optimal bidders despite receiving more profit (see Figure 19). Hence, it can benefit the contractor to select the sharing ratio that benefits the DAA, both in terms of profit per contract and total long-run profit. Figure 19 shows a stark difference between the profit earned by optimal bidders and that earned by the non-optimal bidder. Figure 19(a) depicts that contractor four chooses contract      parameters aggressively; the non-optimal bidding strategy demands more profit per contract but wins far less often, as shown in Figure 19(c). Figure 19(b) shows that in the long run, the optimal bidders receive more total profit because of their higher winning percentages. Contractor four, the non-optimal bidder, would benefit from using the Nash equilibrium bidding strategy rather than aiming for a fixed profit rate. Changing the control parameter limits for the optimal bidders also affects long-run profit. Note that the "cooperative" contractor three demand more profit per contract than the other optimal bidders in Figure 19(a) but wins more contracts according to Figure 19(c). This result is feasible because the DAA and contractors are not playing a zero-sum game, that is, PCF G and PCF K i do not sum to zero.

Integration of PWGEs and AWGEs
The objective for the development of DAA-PWGE, DAA-AWGE, KTR-PWGE and KTR-AWGE analytical and simulation models is to assist the DAA to understand the contractor's perspective and to seek optimum PTB solution and corresponding acquisition strategy under both Government's and contractor's perspectives. Thus, the government PTB solution for a given set of "requirements" should be optimized to achieve government saving and at the same time to have more than one contractor bidding the solution. This means that there will be at least two contractors converge to the same Government's PTB solution with similar market and technology risks as predicted by the DAA-PWGE. And, the contract type and associated contract parameters and incentives are derived based on the compromised results obtained from both DAA-AWGE and KTR-AGWE. In another word, the PTB solution will be obtained from the integrated DAA-PWGE and KTR-PWGE. The final acquisition strategy for acquiring the PTB solution obtained from the integrated DAA-PWGE and KTR-PWGE is to be generated from the integrated DAA-AWGE and KTR-AWGE. As presented in the proposed UGAF shown in Figure 3, the selection of PTB solution and acquisition strategy is a close-loop process and the PTB solution and associated contract type are converge to a single PTB solution with more than one contractor is willing to bid on it. As mention in Section 5, the PTB solution is selected based on the highest PCF score with an assigned "belief" score (or probability). The integrated DAA-PWGE and KTR-PWGE algorithm searches for the highest PCF score and assigned "belief" score that both DAA and KTR can converge to these scores. It is straight forward when the KTR's PCF and "belief" scores converge to DAA's scores. However, when the convergence is not a clear-cut case, a Decision Support Algorithm (DSA) is required to select the optimum PTB solution that can satisfy multiple criteria, including requirements, risks and cost. Our team is currently working on the development of a DSA that leveraged the work done presented in [14][15][16][17][18].
Figures 20 and 21 describe the integration algorithm for integrating DAA-AWGE and KTR-AWGE. The integrated DAA-AWGE and KTR-AWGE algorithm searches for the right "balance" between the DAA's acquisition strategy and KTR's bidding strategy. Our team is currently investigating advanced decision support algorithms that incorporated supervised learning and artificial intelligence to achieve a balance between DAA's and government's perspectives in terms of the government saving and contractor profit, which leads to an estimate the "Expected KTR Profit and Incentives." The "KTR Profit and Incentives" should be compromised with the Government Target Price and associated Target Profit and sharing ratios.

Conclusion and way-forward
This chapter provides an overview of the description of PTB optimization games and acquisition-bidding games from both the government and the contractor perspectives. It presents the DAA-PWGE, DAA-AWGE, KTR-PWGE and KTR-AWGE analytical and simulation models. It also provides flow diagrams to show the combination of these models with the War-Gaming Applications for Achieving Optimum Acquisition of Future Space Systems http://dx.doi.org/10.5772/intechopen.71044 Monte Carlo simulation to generate (i) optimum PTB solutions that can achieve affordability from government perspective, and (ii) optimum FFP and FPIF contract parameters that can achieve affordability from the government's affordability perspective and maximum profit from the contractor perspective. The models were implemented in a collection of MAT-LAB packages.
Simulation results reveal how contractor behavior affects contractor profit. They show that non-optimal bidders demand more profit per contract which results in a lower winning percentage and less total profit in the long run. Therefore, it is beneficial for contractors to implement the Nash equilibrium bidding strategy. In addition, the CPIF model further separates optimal bidders into cooperative, average, and non-cooperative strategies. The analysis shows that contractors can select a less profitable sharing ratio and in turn increase their long run total profit by cooperating with the government. The resulting information can serve as DAA negotiation tools to encourage cooperative bidding in order to increase government savings. The chapter also discusses the integration of contract parameters from the government and contractor perspectives to generate a set of optimum contract parameters that can achieve the "Increased in Competition." The discussion is at high-level and the subject on the selection of the optimum target price and contractor sharing ratios for the FPIF/CPIF contract types to meet the "Increased in Competition" criterion and the application of supervised learning and artificial intelligence to improve the decision-making process deserve more attention for the future research.
The purpose of the unified framework is to set-up a multi-stakeholder acquisition strategy that encourage cooperative bidding leading to a win-win Nash equilibrium. In such a framework, we seek to change the perspectives of the players from antagonistic to collaborative. Since the simulation results show encouraging evidence that our unified war-gaming framework could help achieve the objective of DII, one of our future research directions is to look into the design of a distributed group decision and negotiation systems that would provide an seamless integration of various acquisition models in such a way that an global optimum solution can be found without negatively affecting local solutions. More importantly, the ultimate goal is to fine-tune our generalized model to help involved parties continuously explore new innovative solutions to meet war fighting needs of the digital age.