## 3 Results and Discussion
### Demonstration of Molecular Game-Theoretical System  
Game theory is an effective and powerful tool for analyzing strategic interactions between two or more agents or players. The prisoner's dilemma, a game theory classic, is a paradox in decision analysis in which acting for individual interests is pitted against acting for collective advantage. Generally, each player in the game chooses either \"silence\" or \"betrayal\" [21]. If both players are silent, they both receive the reward R. If both players betray, they both receive the punishment payoff P. If one betrays and the other keeps silent, the defector receives the temptation payoff T, while the silencer receives the \"sucker's\" payoff S. The rank order of payoffs (Figure 4A) is usually T \(>\) R \(>\) P \(>\) S. Because pursuing individual reward logically leads both players to betray, they would receive a better reward if both remain silent as 2R \(>\) T + S.  
With the development of game theory in biology (especially at the cellular level), there are some predictions that game theory could be useful in the interpretation of molecular systems [1; 6]. To extend game-theoretic treatment to an abiotic situation, our above-mentioned constructed artificial molecular system is a successful example of the application of game theory (prisoner's dilemma) to interactions among simple molecules, which relies on the hairpin structure of T33, specifically induced by Hg\({}^{2+}\) and fluorescence quenching, and the molecular adsorption capacities of CoOOH nanosheets [37], resulting in fluorescence intensity and a distribution change in T33 (Figure 4B). From a game theory perspective, the molecular interaction of the \"bait\" molecule T33 with CoOOH and Hg\({}^{2+}\) can be considered a \"game\" in which CoOOH and Hg\({}^{2+}\) have two possible states: absence (silence) or presence (betrayal) regarded as two possible strategies, resulting in conflicts (sharing or self-interest) of the fluorescence distribution of T33, which conforms to the prisoner's dilemma. If both were absent (silent) and the bait molecule T33 did not interact with them, exhibiting its full fluorescence in which that payoff of the whole system was represented as 1, and then each molecular player was equivalent to share the average payoff R = 0.5. If one was absent (silent) and the other was present (betrayal), the existing one selfishly interacted with T33 to form the complex, resulting in fluorescence quenching. That is, in the presence (betrayal) of CoOOH and absence (silence) of Hg\({}^{2+}\), CoOOH selfishly owned the whole system's payoff T + S = 0.67 (note that T = 0.67, S = 0). Payoff S of Hg\({}^{2+}\) was zero. Similarly, in the presence betrayal\") of Hg\({}^{2+}\) and absence (silence) of CoOOH, Hg\({}^{2+}\) selfishly owned the whole system's payoff T + S = 0.96 (note that T = 0.96, S = 0), payoff S of CoOOH was zero. If both were present (betrayal), both competitively interacted with T33 to form T-Hg\({}^{2+}\)-T-CoOOH ternary complex, resulting in stronger quenching of T33. Each molecular player received punishment (payoff P = 0.15), the payoff of the whole system was lowest (2P = 0.3). In this molecular game-theoretical system, the payoff of each molecular player is determined not only by its own state, but also by the state adopted by the molecular coplayers.  
Figure 4: (**A**) Generalized payoff matrix for prisoner’s dilemma where the two players are represented by red and black, and each player chooses to either silence or betrayal; (**B**) Realized payoff matrix for the molecular interaction of the bait molecule T33 with two molecular players, CoOOH and Hg\({}^{2+}\), with different states (absence = silence and presence = betrayal) regarded as strategies, resulting in conflicts (sharing or self-interest) of fluorescence distribution change of T33, which conforms to the prisoner’s dilemma.