Game Theory: Prisoner's Dilemma

• Game Theory mathematically formalizes strategic decision-making in multi-agent interactions.
• Game Theory assumes agents act rationally in their own best interests, with applications in economics, political science, computer science, and biology.

Prisoner's Dilemma

• Two suspects are arrested and cannot communicate, being offered deals by the police.
• If one confesses, they go free while the other gets 10 years.
• If both confess, they each get 5 years.
• If neither confesses, they each get 1 year.

Payoff Matrix Example

• The dilemma illustrates that individual rationality leads to both suspects confessing.
• This leads to a collectively irrational outcome where both are worse off than if they had stayed silent.

Algorithmic Game Theory

• Combines game theory and algorithm design.
• It addresses computational issues in game theory and uses game-theoretic ideas for better algorithm design.
• Key topics include computing equilibria, mechanism design, and the price of anarchy.

Nash Equilibrium

• Defined as a set of strategies where no player benefits from unilaterally changing their strategy.
• A strategy profile $s = (s_1, s_2,..., s_n)$ is a Nash equilibrium if for every player $i$ and every alternative strategy $s'i$, $u_i(s_i, s{-i}) \geq u_i(s'i, s{-i})$
  • $s_i$ is the strategy of player i
  • $s_{-i}$ is the vector of strategies of all players except i
  • $u_i$ is the utility function of player i.

Nash Equilibrium Example: Rock-Paper-Scissors

• Players use a mixed strategy, randomizing their actions.
• The Nash Equilibrium involves each player choosing each action with a probability of 1/3.

Mechanism Design

• Involves designing game rules to achieve desired outcomes despite players' selfish behavior.
• Examples include auction design to maximize revenue, and voting systems to elect the best candidate.

VCG (Vickrey-Clarke-Groves) Mechanism

• Designed to incentivize players to reveal their true preferences.
• It selects the outcome that maximizes the sum of the players' utilities, ensuring social welfare maximization.

Price of Anarchy (PoA)

• Measures the degradation of social welfare due to selfish behavior.
• $PoA = \frac{\text{Social Welfare in the Worst Nash Equilibrium}}{\text{Optimal Social Welfare}}$
• A higher PoA indicates greater inefficiency.

Braess's Paradox

• Shows that adding a road to a network can increase travel time for everyone.
• Selfish routing by drivers can result in an equilibrium worse than if some roads were closed.

Citric Acid Cycle

• Starts with 2 Citrate(p) and 2 Citrate(g), ends with 1 oxaloacetate(p) and 2 oxaloacetate(g)
• Oxidizers:
  • 3 NAD+(p) 1 FAD(p) becomes 3 NADH(p) 1 FAD(p)
  • 6 NAD+(g) 2 FAD(g) becomes 6 NADH(g) 2 FAD(g)
• Waste Products:
  • 2 CO2(p)
  • 4 CO2(g)
• Energy Production:
  • 1 ATP(p)
  • 2 ATP(g)
• Krebs cycle requires oxygen because it is aerobic.

Oxidative Phosphorylation

• Same as the Electron Transport Chain (ETC)
• 32 to 34 total ATP are produced at the end of the ETC per glucose molecule.
• 36-38 total ATP are produced per molecule of glucose in cellular respiration.

ATP per Stage of Cellular Respiration

• Glycolysis: 2 ATP
• Transition: 0 ATP
• Krebs Cycle: 2 ATP
• Electron Transport Chain: 32-34 ATP
• Total: 36-38 ATP

Purpose of ETC

• To make the most ATP without using cellular energy (ATP).
• To use electrons for energy.