# Quantum Conditional Strategies and Automata for Prisoners' Dilemmata under the EWL Scheme
## 3 Definitions and Background
### Classical PD  
Having defined the strategic form of a game in Section 3, we proceed to define the Prisoner's Dilemma (or PD) game. PD is arguably the most famous game in game theory. The traditional story tells of two criminals who have been arrested and are interrogated separately, and subsequently they are offered two options, to either defect or cooperate. Hence, the PD consists of:  
* **A set of players \(N\)**, where \(N=\{1,2\}\);
* for each player \(i\), there is a set of actions \(S_{i}=\{C,D\}\), where \(C\) stands for \"cooperate\" and \(D\) stands for \"defect\";
* the notion of **strategy profiles** captures the combinations of players' strategies such as (DC), (CC), which represent all the combinations of actions that can arise based on the rules of the game. In turn, the actions of the players lead to the payoff utilities assigned to each one of these. Each player's aim is to maximize their own payoff.  
The ordering of the strategy profiles, from best to worst, where the first action in parentheses represents player 1's action and the second action represents player 2's action, is:  
* \((D,C)\), meaning that player 1 defects and player 2 cooperates (resulting in \((t,s)\) payoff);
* \((C,C)\), where both 1 and 2 cooperate (resulting in \((r,r)\) payoff);
* \((D,D)\), where both 1 and 2 defect (resulting in \((p,p)\) payoff), and;
* \((C,D)\), where 1 cooperates and 2 defects (resulting in \((s,t)\) payoff).  
Payoffs in the prism of game theory are numbers (symbolic values can be also used) that reflect the players' motivations and expectations. Payoff values often represent the expected profit, loss, or a utility of the players. They can also stand for continuous measures, or they may express a rank of preference among possible outcomes. Players' payoffs are often represented using a matrix form (like Table 1), but other types of depictions also exist (like the tree form shown in Figure 1) [25; 60].  
The players' actions in the prisoner's dilemma are represented according to the payoff matrix in Table 1[2; 25]. The symbols used are not chosen at random: \(r\) stands for reward, \(s\) for sucker's payoff, \(t\) represents the temptation payoff, and \(p\) is for punishment. The numerical values used later are in accordance with those of [PERSON] et al. in [2].  
If both players cooperate, they both receive the reward \(r\), whereas if they both defect, they receive the punishment \(p\). If player 1 defects while player 2 cooperates, then player 1 gets the temptation payoff \(t\), while player 2 gets the \"sucker's\" payoff \(s\). Symmetrically, if 1 cooperates while 2 defects, then 1 receives the sucker's payoff \(s\), while 2 receives the temptation payoff \(t\). Considering that the players are rational and care only about their individual payoff, \"defect\" is the dominant strategy of the game [25]. For the payoff values, it holds that \(t>r>p>s\), which is necessary for mutual cooperation to be superior to mutual defection for the players, and at the same time defect is guaranteed to be the dominant strategy for both players. The extensive form of the game is described using a game tree, depicted in Figure 1.