# A Game-Theoretical Approach to Clinical Decision Making with Immersive Visualisation
## 3 Proposed Method  
### Game Theory  
Game theory offers mathematical techniques for analysing the dynamics of conflict and cooperation among rational, intelligent decision-making agents [30]. It studies the strategies of interaction among self-interested agents who strive to maximise their gains, specifically strategies of competition, cooperation, negotiation, and coordination [22; 31; 32]. These agents will strategically consider the actions of others and make choices based on those anticipations to maximise their payoff. Central to game theory is the assumption that all involved agents act rationally and prioritise their self-interest. It provides analytical methods to model and analyse system behaviour, such as Nash Equilibrium, Social Optimality, and Pareto optimality [22; 32]. For any non-cooperative game model, each game encompasses, at minimum, a trio of fundamental components. Firstly, there exists a set of agents, colloquially known as players, who participate in the game. Secondly, each agent is endowed with a set of feasible actions that each agent can take. Lastly, there is a set of utility functions assigned to the agents. These utility functions play a critical role in mapping an action profile to real-valued numbers, serving as a quantifiable representation of the individual agent's payoff within the game. Formally, a game \(G=\big{(}P,\{A_{i}\}_{i\in P},\{u_{i}\}_{i\in P}\big{)}\) is defined as a tuple, where  
* \(P=\{1,2,\ldots,n\}\) is a finite set of agents. * For each agent \(i\in\mathrm{P}\), \(A_{i}\) is a set of possible actions (strategies). The set of all joint actions, called strategy profiles, is denoted by \(\mathbf{A}=A_{1}\times A_{2}\times\ldots\times A_{n}\). Each vector \(\mathbf{a}=\left(a_{1}\times a_{2}\times\ldots\times a_{n}\right)\in\mathbf{A}\) is called an action profile. * For each agent \(i\in\mathrm{P}\), \(u_{i}\): \(\mathbf{A}\mapsto\mathbb{R}\) is a utility function that maps an action profile to a real number, indicating the player's payoff for each possible joint action. In game theory, a game can usually be represented in a normal form as an n-dimensional matrix [32; 33]. We use such a matrix in our application of game theory. Each agent is driven to maximise their payoff in a game without knowing other agents' actions. However, due to shared resources and the interdependent nature of the game, an agent's payoff is influenced by both their strategy and those employed by their opponents. In the context of a multiagent system, mixed strategies is defined as follows. Let \(\left(P,\left\{A_{i}\right\}_{i\in\mathrm{P}},\left\{u_{i}\right\}_{i\in \mathrm{P}}\right)\) be a normal-form game, and for any given non-empty set \(X\), let \(\Pi(X)\) be a probability distribution over \(X\). Then a mixed strategy of player \(i\) is an element of  
\[S_{i}=\Pi(A_{i}) \tag{1}\]  
Consequently, the set of mixed-strategies profiles is the Cartesian product of the individual mixed-strategy sets \(S_{1}\times\ldots\times S_{n}\)[32]. The expected utility \(u_{i}\) for agent \(i\) of a mixed-strategy profile \(s=\left(s_{1},\ \ldots,\ s_{n}\right)\) can be calculated as  
\[u_{i}(s)=\sum_{a\ \in\ \Lambda}\ u_{i}(a)\ \prod_{j=1}^{n}s_{j}(a_{j}) \tag{2}\]  
The Nash Equilibrium concept is widely recognised as one of the key principles in game theory. A Nash Equilibrium refers to an action profile where, if any agent were to deviate from their chosen action, they would not achieve a better payoff. As a result, no agents have the incentive or motivation to deviate from such a Nash Equilibrium. Formally, Nash Equilibrium is defined as follows [32]: A strategy profile \(s^{*}\) is a Nash Equilibrium if, for all agents \(i\), and for all strategy \(s_{i}\in S_{i}\)  
\[u_{i}(s^{*}_{i})\ \geq\ u_{i}(s_{i},s^{*}_{-i}) \tag{3}\]  
where \(\left(s_{i},s^{*}_{-i}\right)=\left(s^{*}_{1},\ \ldots,\ s^{*}_{i-1},\ s_{i}\ s^{*}_{i+1}, \ \ldots,\ s^{*}_{n}\right)\) is a strategy profile that is the same as \(s^{*}\) except player \(i\)'s strategy is replaced by \(s_{i}\).