## 6 Experiments
### Evaluation of a Dynamic Duopoly Game
We now outline the steps used to compute the stationary equilibrium and describe the algorithm used for the dynamic duopoly game. We start by observing that the game has a 2-period cycle, so the computation starts with a value function \(v_{1}:S\to\mathbb{R}\) of firm 1, and \(v_{2}:S\to\mathbb{R}\) of firm 2. The payoff function of firm 2 is given by  
\[\pi_{2}(\tilde{a}_{1},a_{2})+\delta\sum_{s_{\omega}}\tilde{o}_{2} (s_{\omega})q(s_{\omega};\tilde{a}_{1},a_{2})\] \[=[(10-\tilde{a}_{1}-a_{2})a_{2}-a_{2}-1]+\delta\sum_{s_{\omega}= \tilde{a}_{2}-0.5}^{s_{\omega}=a_{2}}v_{2}(s_{\omega})q(s_{\omega};\tilde{a}_ {1},a_{2}). \tag{13}\]  
Here, \(\tilde{a}_{1}\) is the state variable that gives the current output of firm 1. **STEP 1:** In this step, firm 2, the player in period 2, chooses \(a_{2}\) to maximize the payoff function \(\pi_{2}(.,.)\). We then compute the value functions of firm 1 and firm 2 using this optimal choice of firm 2. This then gives the value function \(v_{1}^{2}:S\to\mathbb{R}\) of firm 1 for period 2. The payoff function of firm 1 in period 1 is then given by  
\[\pi_{1}(\tilde{a}_{2},a_{1})+\delta\sum_{s_{\omega}}v_{1}^{1}(s_{ \omega})q(s_{\omega};\tilde{a}_{2},a_{1})\] \[=[(10-\tilde{a}_{2}-a_{1})a_{1}-a_{1}-1]+\delta\sum_{s_{\omega}= \tilde{a}_{1}-0.5}^{s_{\omega}=a_{1}}v_{1}^{2}(s_{\omega})q(s_{\omega};\tilde {a}_{2},a_{1}). \tag{14}\]  
In period 1, firm 1 chooses \(a_{1}\) to maximize this payoff function. The value functions obtained by using the optimal \(a_{1}\) then give the value function of both firms 1 and 2, for period 1. In this step, we thus obtain the value functions for both periods 1 and 2 for both firms 1 and 2. However, these value functions are derived by assuming the arbitrary value function \(v_{l}:S\to\mathbb{R}\) for period 2. We, therefore, designate the value functions obtained in this first iteration as \((v_{l}^{1})^{1}:S\to\mathbb{R}\) and \((v_{l}^{2})^{1}:S\to\mathbb{R}\). **STEP 2:** We now go to the second iteration or start the backward induction process from period 4. The value function for period 4 is assumed to be \(\tilde{o}_{l}:S\to\mathbb{R}\). In periods 4 and 3, the analysis is the same as in step 1, and we end up with the value functions \((v_{i}^{2})^{1}:S\to\mathbb{R}\) for period 4, and the value functions \((v_{l}^{1})^{1}:S\to\mathbb{R}\) for period 3. For period 2, we now use the value function \((v_{2}^{1})^{1}:S\to\mathbb{R}\) to define the payoff function of firm 2 for period 2. The payoff function of firm 2 for period 2 is thus  
\[\tau_{2}(\bar{a}_{1},a_{2})+\delta\sum_{s_{\omega}}(v_{2}^{1})^{1}(s _{\omega})q(s_{\omega};\bar{a}_{1},a_{2})\] \[=[(10-\bar{a}_{1}-a_{2})a_{2}-a_{2}-1]+\delta\sum_{s_{\omega}=a_{ 2}-0.5}^{s_{\omega}=a_{2}}(v_{2}^{1})^{1}(s_{\omega})q(s_{\omega};\bar{a}_{1}, a_{2}). \tag{15}\]  
We obtain the optimal choice of firm 2 by choosing \(a_{2}\) that maximizes this payoff function. Using this optimal choice, we then obtain the value functions \((v_{i}^{2})^{2}:S\to\mathbb{R}\). The payoff function of firm 1 for period 1 is now given by  
Figure 10:0 Figure 10:1  
Firm 1 chooses \(a_{1}\) to maximize this payoff function. Using this optimal choice of firm 1, we find the value functions \((v_{1}^{1})^{2}:S\to\mathbb{R}\). The value functions for the firms in periods 1 and 2 from this second iteration is thus, \((v_{i}^{1})^{2}:S\to\mathbb{R}\) and \((v_{i}^{2})^{2}:S\to\mathbb{R}\). \(K^{th}\)-ITERATION: For the general \(K^{th}\) iteration, we use the value functions \((v_{i}^{2})^{K-1}:S\to\mathbb{R}\) for firms 1 and 2, to compute the value functions for periods 2 and 1 as in STEP 2. This will generate the value functions \((v_{i}^{2})^{K-1}:S\to\mathbb{R}\) and \((v_{i}^{1})^{K-1}:S\to\mathbb{R}\) for periods 1 and 2. These iterations give us a sequence of value functions for periods 1 and 2 for firms 1 and 2. These sequences of value functions will converge to the stationary value functions \(v_{1}^{*}:S\to\mathbb{R}\) of firm 1 and \(v_{2}^{*}:S\to\mathbb{R}\) for firm 2. **Claim 1**.