added fairness

README.md

@@ -15,4 +15,9 @@ For additional resources, check out the [AGT textbook](https://www.cs.cmu.edu/~s
 
 ## Contributing
 
-The course reader is built around [Jupyter Book](https://jupyterbook.org/en/stable/intro.html). To contribute, it will be helpful to familiarize yourself with (in order of importance): LaTeX, Markdown, Python, Sphinx, and some basic HTML. 
+The course reader is built around [Jupyter Book](https://jupyterbook.org/en/stable/intro.html). To contribute, it will be helpful to familiarize yourself with (in order of importance): LaTeX, Markdown, Python, Sphinx, and some basic HTML. 
+
+To compile the book locally run:
+```bash
+jupyter-book build docs
+```

docs/_toc.yml

@@ -17,6 +17,7 @@ chapters:
 - file: notes/lec11_crypt
 - file: notes/lec12_pos
 - file: notes/lec13_security_markets
+- file: notes/lec14_fairness
 
 
 

docs/notes/lec14_fairness.md

@@ -0,0 +1,194 @@
+# Envy and Fairness
+
+Consider cake cutting: Alice and Bob want to split a cake, and the cake
+has different toppings (strawberries, chocolate, etc.) over different
+regions. One canonical mechanism is I-cut-you-choose:
+
+1.  One agent cuts the cake;
+
+2.  The other picks which slice they want;
+
+3.  First agent gets the remaining cake.
+
+```{prf:proposition}
+I-cut-you-choose is not dominant-strategy incentive compatible for the
+cutting agent but is for the choosing agent.
+```
+
+```{prf:proof}
+ The cutter's optimal strategy depends on the chooser's
+preferences. The chooser directly chooses their outcome. 
+```
+
+Why, has this mechanism been used? There is some notion of fairness in
+this:
+
+```{prf:definition}
+An allocation is envy-free if no agent envies another agent's
+allocation.
+```
+
+```{prf:proposition}
+The I-cut-you-choose mechanism can always lead to an envy-free
+allocation for both players.
+```
+
+```{prf:proof}
+ The cutter can always cut the cake into two pieces that are
+equal in their perspective, so no matter which slice they get the
+allocation is envy-free for them. Then, the choose chooses their
+favorite slice, so it clearly is envy-free for them. 
+```
+
+## Formalizing Fair Allocations
+
+Goal: from a set of items $A$, choose some allocation for each of $n$
+agents. Each agent $i$ has a valuation function
+$v_i: \mathcal{P}(A) \to \mathbb{R}$. Then, there are several measures
+of fairness (defined in this setting):
+
+-   Envy-Free: for all $i,j$ we have $v_i(A_i) \geq v_i(A_j)$;
+
+-   Proportional: for all $i, v_i(A_i) \geq v_i(A)/n$;
+
+-   Equitable: for all $i,j$ we have $v_i(A_i) = v_j(A_j)$;
+
+-   Perfect: for all $i, v_i(A_i) = v_i(A)/n$.
+
+Another possibility: competitive equilibrium from equal incomes, which
+is similar to competitive equilibrium except give everyone a "fake
+dollar" to spend.
+
+In cases with indivisible goods, often none of them can be satisfied. In
+particular, if there is a single good (and both individuals gain
+strictly positive utility from it), none of these can be satisfied. A
+compromise:
+
+```{prf:axiom}
+Maximin Share (MMS)
+
+1.  Each agent $i$ proposes an allocation $A^1$ to all agents;
+
+2.  The MMS condition is $$v_i(A_i) \geq \max_{A^i} \min_j v_i(A_j^i)$$.
+```
+
+Another compromise can be to approximately satisfy these conditions: let
+$\overline{a}$ be the best item and $\underline{a}$ be the worst item.
+Then,
+
+1.  Envy free up to one item:
+    $v_i(A_i) \geq v_i(A_j \setminus \overline{a})$;
+
+    -   Can always be satisfied.
+
+2.  Envy free up to any item:
+    $v_i(A_i) \geq v_i(A_j \setminus \underline{a})$
+
+    -   Open problem for whether or not this can always be satisfied.
+
+3.  $\alpha$-Maximin Share:
+    $v_i(A_i) \geq \alpha \max_{A^i} \min_j v_i(A_j^i)$
+
+    -   Can always be satisfied for $\alpha \leq 3/4$.
+
+4.  Approximate competitive equilibrium from equal incomes: only require
+    approximate equilibrium.
+
+    -   Can always be satisfied but computationally difficult to compute
+
+Finally, we can directly seek to maximize some "fair" measure of overall
+utility. Some examples are:
+
+1.  Utilitarian/Social Welfare: $\max_{A_1,...,A_n} \sum_i v_i(A_i)$;
+
+2.  Nash social welfare: $\max_{A_1,...,A_n} \prod_i v_i(A_i)$;
+
+    -   Middle ground between Utilitarian and Egalitarian.
+
+    -   Competitive equilibrium from equal incomes maximizes Nash social
+        welfare;
+
+3.  Egalitarian/maximin: $\max_{A_1,...,A_n} \min_i v_i(A_i)$.
+
+## Dominant Resource Fairness
+
+Suppose there are $m$ resources with fixed capacities and $n$ users that
+want to run as many identical tasks as possible. Each task demands some
+vector of resources to run. For now, suppose resources are
+infinitesimally small. How can resources be allocated fairly and
+efficiently without transfers?
+
+For any allocation, each user has some fraction of total capacity of
+each resource (given by point-wise division of the user's usage vector
+by the capacity vector). Then, a user's dominant resource is the
+resource they have the highest fraction of usage of. Dominant resource
+fairness requires equal shares of dominant resources.
+
+```{prf:axiom}
+Algorithm to get DRF Allocation
+
+1.  Every agent submits their ratio of demands;
+
+2.  Algorithm computes the maximal allocation subject to DRF.
+```
+
+```{prf:proposition}
+Every user prefers the DRF allocation to just getting $1/n$ of every
+resource.
+```
+
+```{prf:proof}
+ As the allocation is maximal, some resource $j^*$ is exhausted
+so some agent $i^*$ gets at least $1/n$ of $j^*$. Then, each agent has
+at least $1/n$ of their dominant resource. 
+```
+
+```{prf:proposition}
+The DRF mechanism is envy-free.
+```
+
+```{prf:proof}
+ If $i$ envies $j$, then $j$ must have more of every resource so
+in particular, $j$ must have more of $i$'s dominant resource than $i$
+does, a contradiction. 
+```
+
+```{prf:proposition}
+The DRF mechanism is Pareto-optimal and strategyproof.
+```
+
+```{prf:proof}
+ Pareto-Optimal: As the allocation is maximal, some resource ie
+exhausted and no agent can be made happier without making some other
+agent sadder.
+
+Strategyproof: for any misreport, Pareto-optimality of the allocation
+reached by truthful reporting means that less of the dominant resource
+is received. 
+```
+
+## DRF in Practice
+
+Some of the assumptions made during the setup might break:
+
+1.  Tasks are not identical, arrive over time, and depart;
+
+    -   While resources are not exhausted, select user $i$ with minimal
+        dominant share and add their next feasible task. Not
+        strategyproof (can clump tasks together) but works well in
+        practice.
+
+2.  Tasks might have flexibility in resource use;
+
+3.  Tasks might have more specific criteria;
+
+4.  Tasks not infinitesimally small;
+
+5.  Users might have different priority levels;
+
+    -   Weighted DRF.
+
+6.  Some tasks have zero demand for some resources.
+
+    -   Find the maximal DRF allocation and then recurse on remaining
+        feasible tasks.