# A Data-Driven Game Theoretic Strategy for Developers in Software Crowdsourcing: A Case Study
## 3 Research Design
### The Competition is Strong  
When the competition was strong, the crowd workers' individual interests in maximization turns to be a game problem. The optimal strategy can be obtained by constructing a two-player game model, within the context of complete information and solving the mixed-strategy Nash equilibrium. Same as for the above, before we discuss the reward strategy, firstly we carry out the reasonable assumptions and describe the question. **Assumption 2. **_When the competition was strong, there was a certain probability that worker A and worker B, both with a strong crowdsourcing ability, can got rewarded. At the same time, they knew each other very well (in fact, the workers can communicate with each other on the Topcoder.com community), and the crowdsourcing tasks were relatively abundant in this period (in fact, crowdsourcing tasks are added dynamically)._  
**Question 2. **_When worker A and worker B are faced with the same crowdsourcing task, their winning probabilities are \(P_{a}\) and \(P_{b}\), respectively, the times needed to be consumed are \(t_{a}\) and \(t_{b}\) respectively, and the reward they get is \(\mathds{R}\). In fact, the reward is divided into two forms: payment (the top is \(P\) and the second is \(p\)) and score (\(S\)). The distribution of the score is adjusted by the parameter (in fact Topcoder.com sets it as 0.7, that is, the top worker gets a \(70\%\) score and the second gets the rest). What are the probabilities of worker A and worker B getting the maximum reward if they take part in the same task?_  
_Formal description:_ In a game \(\mathds{G}=(S_{1},\cdots,S_{n};u_{1},\cdots,u_{n})\) of a crowdsourcing task participated by \(n\) developers, the mixed-strategy profile constructs a \(p^{*}=\left\{p_{1}^{*},\cdots,p_{i}^{*},\cdots,p_{n}^{*}\right\}\) Nash Equilibrium, for all \(i=1,2,\cdots,n\) the formula below holds:  
\[v_{i}\big{(}p_{i}^{*},p_{-i}^{*}\big{)}\geq v_{i}\big{(}p_{i},p_{-i}^{*}\big{)},\forall p_{i}\in\sum_{i}p^{*} \tag{10}\]  
The strategy forms a Nash equilibrium if one strategy group makes the strategy of any participant the best compared to other participants' strategies. Here we focus on the analysis of the two-player game when workers choose the task at Topcoder.com. The pay-off matrix constructed with complete information is shown in Table 2.  
In Table 2, \(\pi(A_{11})\) stands for [PERSON]'s reward and \(\pi(B_{11})\) for [PERSON]'s reward when both worker [PERSON] and [PERSON] participate; \(\pi(A_{10})\) stands for [PERSON]'s reward when [PERSON] is in the contest while [PERSON] is not, and \(\pi(B_{10})\) represents [PERSON]'s reward when [PERSON] is in while [PERSON] is not. Each reward is calculated as follows:  
\[\pi(A_{11})=P_{a}(P+\alpha S)+(1-P_{a})(p+(1-\alpha)S)-t_{a} \tag{11}\]  
\[\pi(B_{11})=P_{b}(P+\alpha S)+(1-P_{b})(p+(1-\alpha)S)-t_{b} \tag{12}\]  
\[\pi(A_{10})=P+S-t_{a} \tag{13}\]  
\[\pi(B_{10})=P+S-t_{b} \tag{14}\]  
After the calculation of each reward, the mixed Nash Equilibrium in this game is figured out. The reward functions of the worker A and worker B are shown as below, respectively:  
\[E_{A}(\mu,\theta)=\mu[\theta(\pi(A_{11}))+(1-\theta)(\pi(A_{10}))] \tag{15}\]  
\[E_{B}(\mu,\theta)=\theta[\mu(\pi(B_{11}))+(1-\mu)(\pi(B_{10}))] \tag{16}\]  
As reward is calculated in this way: [PERSON]'s reward when both A and B participate plus [PERSON]'s reward when A participate while B does not. After these calculations are done, the next step is to work out \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(B_{1}(\theta)\) & \(B_{0}(1-\theta)\) \\ \hline \(A_{1}(\mu)\) & \((\pi(A_{11}),\pi(B_{11}))\) & \((\pi(A_{10}),\pi(A_{10}),0)\) \\ \(A_{0}(1-\mu)\) & \((0,\pi(B_{10}))\) & \((0,0)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pay-off matrix. and \(\theta\). Here we use the method of payment maximization to solve the Nash Equilibrium of the game.