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# Algorithmic Game Theory ## What is Game Theory? Game theory is the study of mathematical models of strategic interactions among rational agents. It has applications in many fields, including: * Economics * Political Science * Biology * Computer Science In game theory, we study the beha...

# Algorithmic Game Theory ## What is Game Theory? Game theory is the study of mathematical models of strategic interactions among rational agents. It has applications in many fields, including: * Economics * Political Science * Biology * Computer Science In game theory, we study the behavior of agents in strategic settings, where the outcome of an agent's actions depends on the actions of other agents. ### Example: Prisoner's Dilemma Two suspects are arrested for a crime. The police do not have enough evidence to convict them, so they offer each suspect a deal: * If one suspect confesses and the other does not, the confessor goes free and the other gets 10 years in prison. * If both confess, they each get 5 years in prison. * If neither confesses, they each get 1 year in prison. This can be represented in a payoff matrix: | | Suspect B Confesses | Suspect B Stays Silent | | :---------- | :------------------ | :--------------------- | | **Suspect A Confesses** | -5, -5 | 0, -10 | | **Suspect A Stays Silent** | -10, 0 | -1, -1 | ## Algorithmic Game Theory Algorithmic game theory is an area that lies at the intersection of game theory and algorithm design. ### Motivation * **Classical game theory** often ignores the computational aspects of games. * **Real-world games** are often too complex to be solved by hand. * **Algorithms** can be used to find or approximate game-theoretic solutions. ### Topics in Algorithmic Game Theory * **Mechanism Design:** How to design games with desirable properties. * **Price of Anarchy:** How much efficiency is lost due to selfish behavior. * **Computation of Equilibria:** How to find Nash equilibria and other solution concepts. ## Example: Network Routing Consider a network with $n$ users who want to route traffic from a source $s$ to a destination $t$. Each user can choose a path from $s$ to $t$, and the cost of a path depends on the amount of traffic on that path. Each user wants to minimize their own cost, but their choices affect the costs of other users. This is a game! **Question:** How does the overall efficiency of the network degrade due to the selfish behavior of the users? This is an example of a "price of anarchy" question.