Algorithmic Game Theory

Algorithmic Game Theory

• Combines game theory and algorithm design.

Game Theory

• The study of strategic interactions among rational agents.

Algorithm Design

• Focuses on designing algorithms to achieve specific objectives.

Importance

• Traditional game theory assumes computationally unbounded players, which is often not the case in real-world applications.
• Modern applications require consideration of strategic participants and computational efficiency.
• Examples = Online Ads, Spectrum and Search Auctions, Network Routing

Selfish Routing Model

• A network includes n nodes and m edges.
• Each edge e has a cost function $l_{e}(x)$, representing the cost per unit of traffic when traffic is x.
• k sets of players (commodities) exist.
• For each set i, $r_{i}$ players want to move one unit of traffic from $s_{i}$ to $t_{i}$.

Flow Definition

• A flow f assigns a rate of traffic on each edge, satisfying:
• For each commodity i, the flow leaving $s_{i}$ and entering $t_{i}$ is $r_{i}$.
• For each node (excluding source and sink), inflow equals outflow.
• The cost of flow f is the average latency experienced, calculated as $C(f) =\sum_{e} l_{e}(f_{e}) \cdot f_{e}$, where $f_{e}$ is the flow on edge e.

Nash Equilibrium

• A flow f is at Nash equilibrium when no player can reduce their cost by unilaterally changing paths.

Price of Anarchy (PoA)

• Measures inefficiency, the ratio of the worst Nash equilibrium cost to the social optimum cost.
  • $PoA = \frac{\text{Cost of worst Nash equilibrium}}{\text{Cost of social optimum}}$
• Social optimum is the flow minimizing total cost.

Braess's Paradox Example

• A four-city network ($A$, $B$, $C$, $D$) has two routes from $A$ to $D$: $A \rightarrow B \rightarrow D$ and $A \rightarrow C \rightarrow D$.
• One unit of traffic goes from $A$ to $D$.
• Cost functions of the edges:
  • $l_{AB}(x) = x$
  • $l_{BD}(x) = 10$
  • $l_{AC}(x) = 10$
  • $l_{CD}(x) = x$

Braess's Paradox Without Dashed Edge

• Nash equilibrium involves half the traffic taking each route ($A \rightarrow B \rightarrow D$ and $A \rightarrow C \rightarrow D$).
• Cost per player is $0.5 + 10 = 10.5$.
• Total cost is $10.5$.

Braess's Paradox With Dashed Edge

• A zero-cost edge is added from $B$ to $C$.
• Now, everyone takes the path $A \rightarrow B \rightarrow C \rightarrow D$.
• The cost per player is $1 + 0 + 1 = 2$.

Bounding the Price of Anarchy Theorem

• If all cost functions are linear, i.e., $l_{e}(x) = a_{e}x + b_{e}$ for some $a_{e}, b_{e} \geq 0$, then the price of anarchy is at most $\frac{4}{3}$.

Bounding the Price of Anarchy Proof

• Let $f$ be a Nash equilibrium flow and $f^{*}$ be a social optimum flow.
• $C(f) = \sum_{e} f_{e}l_{e}(f_{e}) = \sum_{e} f_{e}(a_{e}f_{e} + b_{e})$
• $C(f^{}) = \sum_{e} f_{e}^{}l_{e}(f_{e}^{}) = \sum_{e} f_{e}^{}(a_{e}f_{e}^{*} + b_{e})$
• Since $f$ is a Nash equilibrium, for any $s_{i}-t_{i}$ path $P$ with flow $f_{P} > 0$, we have: $\sum_{e \in P} l_{e}(f_{e}) \leq \sum_{e \in P'} l_{e}(f_{e})$
• For any other $s_{i}-t_{i}$ path $P'$ with flow $f_{P'}^{} > 0$ in the optimal flow $f^{}$, we have: $\sum_{e \in P} l_{e}(f_{e}) \leq \sum_{e \in P'} l_{e}(f_{e})$
• Therefore: $\sum_{e} f_{e}^{} l_{e}(f_{e}) \leq \sum_{e} f_{e}^{} l_{e}(f_{e}^{*})$
• $C(f) = \sum_{e} f_{e}l_{e}(f_{e}) = \sum_{e} f_{e}(a_{e}f_{e} + b_{e}) = \sum_{e} a_{e}f_{e}^{2} + \sum_{e} b_{e}f_{e}$
• $\leq \sum_{e} a_{e}f_{e}^{2} + \sum_{e} b_{e}f_{e} + \sum_{e} f_{e}^{} (a_{e}f_{e} + b_{e}) = \sum_{e} a_{e}f_{e}^{2} + \sum_{e} b_{e}f_{e} + \sum_{e} a_{e}f_{e}^{}f_{e} + \sum_{e} b_{e}f_{e}^{*}$
• $\leq \frac{4}{3}\sum_{e} a_{e}(f_{e}^{})^{2} + \frac{4}{3}\sum_{e} b_{e}f_{e}^{}$
• $\sum_{e} f_{e}^{} l_{e}(f_{e}^{}) = \sum_{e} a_{e}(f_{e}^{})^{2} + \sum_{e} b_{e}f_{e}^{}$
• $C(f) \leq \frac{4}{3}C(f^{*})$
  • Therefore, the price of anarchy is at most $\frac{4}{3}$.

Matriisilaskenta ja Lineaarialgebra (Matrix Calculus and Linear Algebra)

Tentti (Exam) 24.10.2011

Tehtävä 1 (Task 1)

• Given $\mathbf{A} = \begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix}$.

Ominaisarvot ja Ominaisvektorit (Eigenvalues and Eigenvectors)

• Calculate the eigenvalues and corresponding eigenvectors of matrix $\mathbf{A}$.

Diagonalisointi (Diagonalization)

• Diagonalize matrix $\mathbf{A}$, i.e., find matrices $\mathbf{V}$ and $\mathbf{D}$ such that $\mathbf{A} = \mathbf{V}\mathbf{D}\mathbf{V}^{-1}$.

Tehtävä 2 (Task 2)

Lineaarinen Riippumattomuus (Linear Independence)

• Determine for which values of the constant $a \in \mathbb{R}$ the vectors $\begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$, $\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$, and $\begin{bmatrix} 2 \ a \ 1 \end{bmatrix}$ in vector space $\mathbb{R}^3$ are linearly independent.

Kanta Aliavaruudelle (Basis for Subspace)

• Determine a basis for the subspace of the vector space $\mathbb{R}^3$ spanned by the vectors $\begin{bmatrix} 1 \ 2 \ 1 \end{bmatrix}$, $\begin{bmatrix} 2 \ 1 \ 1 \end{bmatrix}$, and $\begin{bmatrix} 0 \ 3 \ 1 \end{bmatrix}$.

Tehtävä 3 (Task 3)

• Determine all matrices $\mathbf{X} \in \mathbb{R}^{2 \times 2}$ that satisfy the condition $\mathbf{A}\mathbf{X} = \mathbf{X}\mathbf{A}$, where $\mathbf{A} = \begin{bmatrix} 1 & 2 \ -1 & -1 \end{bmatrix}$.

Tehtävä 4 (Task 4)

• Let $\mathbf{A} \in \mathbb{R}^{n \times n}$. Prove that matrix $\mathbf{A}^2$ and matrix $\mathbf{A}$ have the exact same eigenvectors.

New Deal programs

Civilian Conservation Corp (CCC)

• Relief & Recovery
• Description: Provided families and unemployed young man jobs for chance. Living wage Food and shelter. Gaining valuable skills. Reconstruction of wilderness
• Who did it help?: Families, unemployed, Young Men

Public Works Administration

• Recovery
• Description: Unemployment relief. Built dams, schools, bridges Facilitating the country’s infrastructure, Employing for projects
• Who did it help?: Unemployed, All people

Works Progress Administration

• Recovery & Relief
• Description: Highways, airports, schools, buildings,roads. Supported artists, writers, musicians. Healthcare. Diverse range of support and care. Employment funding
• Who did it help?: Families, Employed, Unemployed, Artists, economy, business

Tennessee Valley Authority

• Recovery & Reform
• Description: Source of electricity, Flood control agricultural improvement, Hydroelectric power, Economic growth, Construction of dams.
• Who did it help?: Residents, Tennessee Valley, Business

Federal Deposit Insurance Corporation

• Reform
• Description: Protecting Deposits, Stability in finance, Trust and security, Safety of funds, prevent bank runs, restore public confidence, Supervising and regulating banks
• Who did it help?: Businesses employed, economy