Added Auction Theory notes.

Auction Theory/01 - (left out) Mechanism Design and Game Theory.md

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+(See the slides, I left out this lecture).

Auction Theory/02 - Social Choice, Utility, Mechanism Design and Quasilinear Mechanisms.md

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+# 02 - Social Choice, Utility, Mechanism Design and Quasilinear Mechanisms
+
+Overview of related fields:
+
+- Game theory
+- Social Choice Theory
+- Mechanism Design
+
+
+## Social Choice Theory
+Voting rule: n voters, m alternatives, preference profile for each voter
+
+- Plurality rule
+- Borda rule
+- **Condorcet criterion**: candidate that wins all pairwise majority comparisons (does not always exist)
+	- Condorcet paradox: cycles can occur!
+- Majority criterion (isn't this = plurality?)
+- Consistency: if alternative chosen, it it also chosen from all subsets in which it is contained
+	- Plurality voting is not consistent!
+	- Example: what happens if a candidate dies one day after the election - is the election still valid?
+
+
+### May's Theorem
+"2 Alternatives are easy"
+
+### Social Choice vs. Social Welfare functions
+- $O$: finite set of alternatives/outcomes
+- $L^n$: set of preference orderings of a set $I$ of n voters
+- *Social choice function* over $I$ and $O$: $C: L^n \to O$
+	- produces one alternative (different from the definition in CSC!)
+- *Social welfare function* over $I$ and $O$: $W: L^n \to L$
+	- produces preference ranking
+
+### Arrow's Impossibility Theorem (1950)
+Some principles for a social welfare function:
+- Pareto efficiency
+	- If everybody prefers a to b, a should be preferred to b
+- Non-dictatorship (not one voter s.t. its preferences are always copied)
+- Independence of irrelevant alternatives (IIA)
+	- if a is preferred to b, and then the votes are changed s.t. the relative ordering between a and b stays intact in each vote, then a should still be preferred to b.
+
+
+*Arrow's Impossibility Theorem*: Assume 3 alternatives. A social welfare function which is Pareto efficient and satisfies IIA is dictatorial.
+
+##### Proof sketch
+Start with the situation where everybody prefers A and C to B. (by Pareto efficiency: B ranked last). Now go through to the voters in turn and for each, move B to the top of their preference list, until B is the preferred outcome overall.
+
+Show: there is a pivotal voter who causes the swing, using IAA
+
+Show then: the pivotal voter $i$ dictates society's decision between $A$ and $C$ (local dictator over this set)
+
+(kinda unclear to me... Why does this dictator argument work in general, not just for the particular preference profile we constructed?)
+
+### Social Choice Impossibility results
+... (see slides)
+
+### Manipulability
+All known voting rules are (sometimes) manipulable!
+
+##### Gibbard-Satterthwaite Impossibility Theorem
+Assume at least 3 alternatives. No rule is simultanesouly
+- onto (every alternative can win, in principle)
+- non-dictatorial,
+- non-manipulable (i.e. dominant-strategy truthful)
+
+### Circumventing Impossibilities
+- Relaxing properties
+- randomization
+	- random serial dictatorship is strategy-proof!
+- limiting the domain (e.g. only two candidates; single peaked votes; allow for monetary transfers)
+
+
+## Utility Functions
+Ordinal vs. cardinal measures.
+
+Ordinal: advantage of comparability, but disadvantage that it cannot express intensity. In market environments, cardinal utilities are often reasonable.
+
+### Uncertainty
+Lotteries, etc.
+
+St. Petersburg Paradox: resolved by *diminishing marginal utility*, i.e. $u'(o) > 0$ and $u''(o) < 0$. Referred to as *Bernoulli utility function*.
+
+Simplest example: $U(o) = ln(o)$.
+
+### Axioms of Expected Utility Theory
+Axioms for *Von-Neumann-Morgenstern utility functions*:
+
+A.1 $\succcurlyeq$ is complete
+
+A.2 $\succcurlyeq$ is transitive
+
+A.3 $\succcurlyeq$ is continuous: if $p \succcurlyeq q \succcurlyeq r$, there is an $\alpha$ s.t. $\alpha p + (1-\alpha)r \sim q$
+
+A.4 $\succcurlyeq$ is independent: if $p \succ q$, then for all $\alpha \in (0, 1]$: $\alpha p + (1-\alpha)r \succ \alpha q + (1-\alpha)r$.
+
+$\succcurlyeq$ satisfies the Axioms A1-A4 if and only if there is a utility function $U(o)$ s.t. for all $p, q \in \Delta(O)$, $p \succcurlyeq q \Leftrightarrow U(p) \geq U(q)$.
+
+
+
+## Mechanism Design
+Goal: we'd like to use the private information of each agent to implement a social choice function, and would like to avoid them lying about it.
+
+Example: "outcome 1 gives me 100000000 utility and everything else 0" even if they only slightly prefer outcome 1
+
+
+### Bayesian Mechanism Design
+Bayesian game: $(I, A, \Theta, F, u)$ with
+- *agents* $I$, 
+- *action set* $A = A_1 \times \dots \times A_n$,
+-  *type space* $\Theta = \Theta_1 \times \dots \times \Theta_n$,
+-   common *type prior* $F: \Theta \to [0, 1]$,
+-    *utilities* $u = (u_1, \dots, u_n)$, $u_i: A \times \Theta \to \mathbb{R}$.
+
+For mechanisms: actions not equivalent to outcomes anymore! Introduce
+- $O$ is the set of *outcomes*,
+- $u_i$ now has signature $u_i: O \times \Theta \to \mathbb{R}$
+
+Define the Bayesian game that is being played as $(I, O, \Theta, F, u)$, i.e. with $O$ as action space.
+
+A *mechanism* in such a game is a pair $(A, M)$ where $A$ are the actions and
+- $M: A \to \Delta(O)$
+
+In other words: the players choose their actions, which the center of the mechanism translates to a distribution over the outcomes. These determine the utilities.
+
+##### Implementation of social choice functions
+The mechanism $M$ *implements* a social choice function if there is an equilibrium induced by $M$ that matches the outcome of the social choice function. Formally:
+
+- $M$ *implements $C$ in a dominant strategy* if for any $u$, the game $(I, O, \Theta, F, u)$ has an equilibrium in dominant strategies and in any such equilibrium $a^\ast$, $M(a^\ast) = C(u)$
+- $M$ *implements $C$ in a Bayesian Nash equilibrium* if the Bayesian game $(I, O, \Theta, F, u)$ has a Bayesian Nash equilibrium such that for each $\theta$ action profile $a$ that can arise in the equilibrium given $\theta$, $M(a) = C(u(\cdot, \theta))$
+
+In other words: the choice function selects an outcome according to the preference of the agents, i.e. $u$. 
+
+
+### Design Goals: Incentive Compatibility
+- *Incentive compatibility* (*truthfulness*): No incentive to lie about one's type
+- *Dominant-strategy incentive compatible* (i.e. *strategyproof*) mechanism: $$
+\forall i, \theta_1,\dots,\theta_n, \theta_i':~ u_i(o(\theta_1,\dots,\theta_i,\dots,\theta_n), \theta_i) \geq u_i(o(\theta_1,\dots,\theta_i',\dots,\theta_n), \theta_i)$$
+
+
+
+
+### Direct Revelation Mechanisms
+Observation: there is no need for arbitrarily complicated mechanisms, we can focus on *direct-revelation mechanism* where every bidders reports their type truthfully. => Revelation principle (Gibbard 1973)
+
+##### Revelation principle
+"The agents don't have to lie, because the mechanism already lies for them"
+
+![[direct-revelation.png]]
+
+
+### Single-Peaked Preferences
+Assumption: alternatives are ordered on a line, and each voter has a most preferred alternative, and prefers the other alternatives in order of the distance from their preferred alternative. 
+
+Strategyproof mechanism: choose mediam mechanism. (Also Condorcet winner!)
+
+
+## Quasilinear Mechanism Design
+
+### Quasilinear utility functions
+We separate non-monetary and monetary outcomes:
+- Define *outcome* $o\in O$: o = (x, p) s.t. x is a non-monetary outcome and $p$ is an n-dimensional payment vector of payments to the individual agents
+- agents can valuate the non-monetary outcomes: $v_i(x, \theta_i) \in \mathbb{R}$
+
+A *utility function*  in this context is computed as the utility of the value of the non-monetary outcome minus the payment:
+$$u_i(o, \theta_i) = U_i(v_i(x, \theta_i) - p_i)$$
+
+A utility function is *quasi-linear* if it is "linear in the payment", (but apparently actually any monotonic $U_i$ means that $u_i$ is considered quasi-linear). Now we assume risk-neutrality, i.e. $u_i(o, \theta_i) = v_i(x, \theta_i) - p_i$.
+
+##### Some properties of quasilinear utility functions (?)
+- no budget constraints: amount of money the agent has does not influence utility
+- no externalities: agents don't care how much others have to pay
+- transferable utility: money can be used to transfer utility
+
+
+### Interpersonal Utility Comparisons
+"we assume *interpersonal utility comparisons*" (whatever that means) => allows for 
+- *utilitarian social welfare function*: overall welfare = sum of individual utilities
+- *Rawlesian social welfare function*: overall welfare = utility of least-well-off individual
+
+Assumption for now: utilitarian.
+
+### Mechanism Design Goals
+#### Incentive Compatibility in Quasilinear, utilitarian setting
+Compare with [[02 - Social Choice, Utility, Mechanism Design and Quasilinear Mechanisms#Design Goals Incentive Compatibility | this section]], but now with the quasi-linear utilities substituted.
+
+- *Dominant-strategy incentive compatible* (i.e. *strategyproof*) mechanism: $$
+\forall i, \theta_1,\dots,\theta_n, \theta_i':~ v_i(x(\theta_1,\dots,\theta_i,\dots,\theta_n), \theta_i) - p_i(\theta_1,\dots,\theta_i,\dots,\theta_n) \geq$$ $$v_i(x(\theta_1,\dots,\theta_i',\dots,\theta_n), \theta_i) - p_i(\theta_1,\dots,\theta_i',\dots,\theta_n) $$
+
+- *Bayes-Nash incentive compatible* if telling the truth is a BNE:
+$$\forall i, \theta_k, \theta_i': 
+\sum_{\theta_{-i}} P(\theta_{-i}) (v_i(x(\theta_1,\dots,\theta_i,\dots,\theta_n), \theta_i) - p_i(\theta_1,\dots,\theta_i,\dots,\theta_n)) \geq
+$$
+$$
+\sum_{\theta_{-i}} P(\theta_{-i}) (v_i(x(\theta_1,\dots,\theta_i',\dots,\theta_n), \theta_i) - p_i(\theta_1,\dots,\theta_i',\dots,\theta_n))
+$$
+
+#### Individual Rationality
+Avoid selfish center: "all agents give me all their money" => agents would not participate.
+
+In an individually rational mechanism, no participant can be made worse off if deciding to participate in the mechanism.
+
+- *ex-post individually rational* mechanism: $$\forall i, \theta_1,\dots,\theta_n: ~ v_i(x(\theta_1,\dots,\theta_n), \theta_i) - p_i(\theta_1, \dots, \theta_n) \geq 0$$
+
+
+### The VCG (Vickrey-Clarke-Groves) Mechanism
+Goal: design mechanism which is 
+- efficient (maximize utilitarian SWF)
+- truthful revelation is an equilibrium
+- individually rational
+- budget balanced
+
+i.e. get around [[#Gibbard-Satterthwaite Impossibility Theorem]] using quasi-linear utilites
+
+
+#### VCG mechanism
+(Notation slightly adjusted from the slides)
+
+Mechanism where 
+1. Each $i$ *reports type* $\theta_i'$ and has valuation $v_i'(x) = v_i(x, \theta_i')$ (max amount $i$ would pay the center for choice $x$).
+2. Mechanism *chooses outcome* $x = \arg\max_x \sum_i v_i(x, \theta_i')$
+3. Mechanism *determines payments* of each agent $i$:
+	1. Pretend $i$ does not exist and choose $y = \arg\max \sum_{j\neq i} v_j(y, \theta_j')$
+	2. $i$ must make payment $\sum_{j\neq i} v_j(y, \theta_j') - \sum_{j\neq i} v_j(x, \theta_j')$: The difference between the other agent's total welfare between $y$ and $x$, i.e. how much the existence of $i$ damages the other agents.
+
+The i-th agent has utility $\sum_i v_i'(x, \theta_i') - \max_y \sum_{j \neq i} v_j' (y, \theta_j')$: their *marginal contribution to welfare*.
+
+Some more explanation (taken from [Noam Nisan, Introduction to Mechanism Design (for Computer Scientists)]):
+- the term $- \sum_{j\neq i} v_j(x, \theta_j')$ makes the mechanism incentive compatible, by aligning the i-th bidder's incentives with the goal of maximizing social welfare
+- payments of $h_i(\theta'_{-i}) - \sum_{j\neq i} v_j(x, \theta_j')$ would be incentive compatible for any function $h_i$ that is independent of the i-th agent's reported types
+- one wants to choose $h_i$ in such a way that it is *individually rational* (all players always get non-negative utility), and has *no positive transfers* (no player is ever paid money).
+- these properties are provided by *Clarke's pivot rule*  $h_i(\theta'_{-i}) = \max_y \sum_{j \neq i} v_j(y, \theta'_j)$. Choosing this $h_i$ leads to VCG payments.
+
+#### Vickrey Auction
+The second-price Vickrey auction is a VCG mechanism: $v_i(x, \theta_i)$ is the value the item has if $i$ gets the item in $x$ and 0 otherwise. The bidder therefore gives the item to the highest-bidding agent, who has to pay the second-highest price ($y$ is that the second-highest bidder wins; so the second-highest price minus what the other agents get if $x$ is chosen, i.e. 0)
+
+#### Strategyproofness of VCG Mechanism
+The i-th agent can neither affect the choice of $y$, nor the terms $v_i'(x, \theta_i')$. Maximizing the utility therefore is equivalent to maximizing $v'_i(x, \theta_i)$, which is the case if $\theta_i' = \theta_i$.
+
+=> Truth-telling is a dominant strategy
+
+#### Grove mechanisms
+Green and Laffont 1977, Holmstrom 1979: The general class of Groves mechanisms are the only mechanisms that implement *efficient allocation in dominant strategies with quasi-linear utility functions*.
+
+#### Downsides of VCG
+- trusted center required
+- not always budget-balanced
+- Collusion: Strategy-proof against single actors, but not against groups of actors
+
+
+### Wilson Doctrine
+R. Wilson (1987):
+The deficiency of Bayesian game theory is that the common prior assumption is unrealistic. Common knowledge assumptions need to be weakened to approximate reality.