Week 11: Fear and Loathing on the Pareto Frontier

	Start	End	Topic
Lecture	3:30pm	4:00pm	A Whirlwind Tour of Prisoners’ Dilemmas
	4:00pm	4:50pm	Policy Interventions: Transforming Prisoners’ Dilemmas into Assurance/Invisible Hand Games
Break!	5:00pm	5:10pm
	5:10pm	6:00pm	“Inverse Fairness”(!): Machine-Learning What Policies Value

Where We Left Off…

Prisoners’ Dilemma feels like a silly math/econ problem at first… then you get brainwashed by pol econ PhD and suddenly see it at the “core” of 95% of global issues

Bowles & Jayadev (2014), “One Nation Under Guard”, *NYT*

**Newfoundland: cod landings in tons (1851–2014):** In the 1960s new fishing technologies allowed a dramatic increase in cod fishing, far outpacing the capacity of the fish to reproduce, and leading to total collapse by 1992 when the Canadian government banned fishing entirely. Restoration of fishing stocks to the sustainable levels of the past may occur by the 2030s.

Fishers’ Dilemma (Our “Core” Prisoners’ Dilemma)

Single, unique Nash equilibrium, and it’s Pareto inferior
(looming in background: unsustainable if total hours/day > 14)

The “Iterated Elimination” Result

		\(\color{#e69f00}B\)
		Fish 6h		Fish 8h
\(\color{#0072b2}A\)	Fish 6h	\(\color{#0072b2}\cancel{1}\color{black}, \,\)	\(\color{#e69f00}\cancel{1}\)	\(\color{#0072b2}\cancel{0}\color{black},\)	\(\color{#e69f00}\boxed{\mathbf{1.2}}\)
\(\color{#0072b2}A\)	Fish 8h	\(\color{#0072b2}\boxed{\color{#0072b2}\mathbf{1.2}}\color{black}, \,\)	\(\color{#e69f00}\cancel{0}\)	\(\color{#0072b2}\boxed{\color{#0072b2}\mathbf{0.4}}\color{black},\)	\(\color{#009e73}\boxed{\color{#e69f00}\mathbf{0.4}}\)

Boxes = Best Responses:

\[ \begin{aligned} {\color{#0072b2}\text{BR}_A}({\color{#e69f00}\overset{B}{6\textrm{h}}}) &= {\color{#0072b2}8\textrm{h}}, \; {\color{#0072b2}\text{BR}_A}({\color{#e69f00}\overset{B}{8\textrm{h}}}) = {\color{#0072b2}8\textrm{h}} \\ {\color{#e69f00}\text{BR}_B}( {\color{#0072b2}\underset{A}{6\textrm{h}}} ) &= {\color{#e69f00}8\textrm{h}}, \; {\color{#e69f00}\text{BR}_B}( {\color{#0072b2}\underset{A}{8\textrm{h}}} ) = {\color{#e69f00}8\textrm{h}} \end{aligned} \]

Best response is always \(\text{8h}\), no matter what other player does!

\[ \begin{aligned} \implies &{\color{#0072b2}\mathbb{E}[u_A]} = {\color{#e69f00}\mathbb{E}[u_B]} = 0.4 \text{ for now}, \\ &\leadsto \; ? \text{ once fishery collapses (}\textstyle\sum\text{hrs} = 16\text{)} \end{aligned} \]

Pareto Dominance

Policy Intervention : Allow Contracts

\(\leadsto\) Operationalizing Power as “second best” outside option(s)

Equally good outside options \(\implies\) can contract to Pareto-optimal point \(o^P\)
\(B\) has better outside options \(\implies\) can make take it or leave it offer to \(A\):
- “You (\(A\)) fish 6 hrs all the time. I (\(B\)) fish 6 hrs 41% of time, 8 hrs otherwise”

Slightly better for \(A\) \(\implies\) \(A\) accepts:

\[ \begin{aligned} \mathbb{E}[u_A(a_A = \textsf{Reject})] &= \overset{\text{STTP}}{\boxed{0.4}} \; \text{ (prev slide) } \overset{\text{LTTP}}{\color{#e69f00}\boxed{\color{black}\leadsto -\infty}} \\ \mathbb{E}[u_A(a_A = \textsf{Accept})] &= 0.41\cdot 1 + 0.59 \cdot 0 = \boxed{0.41} \\ \mathbb{E}[u_B(a_A = \textsf{Reject})] &= \boxed{0.4} \; \text{ (prev slide) } {\color{#e69f00}\boxed{\color{black}\leadsto 0.39}} \\ \mathbb{E}[u_B(a_A = \textsf{Accept})] &= 0.41\cdot 1 + 0.59 \cdot 1.2 = \boxed{1.118} \end{aligned} \]

\(B\)’s offer = credible threat in both short and long term; same threat from \(A\) would not be credible (\(B\) knows \(A\) would eventually die: \(u_A \leadsto -\infty\))

HW4: observe policy outcome \(o^{\text{TIOLI}}_{B \rightarrow A}\) \(\Leftrightarrow\) social welfare weights \(\omega_B > \omega_A\)

Policy Intervention : Fines for Overfishing

\(\leadsto\) Weber’s descriptive definition of “The State”: Agent with Monopoly on Legal Use of Force (Weber 1919) (remember him?)

Notice: Previous “intervention” was actually self-enforcing! However, outcome was…
- Determined entirely by asymmetric power, and
- Took no account of anyone in society besides two fishers!
- (Thought experiment: if both had “good” outside options, best for them could be fish cod to extinction then move on to “second-best” option \(u = 0.4 \leadsto 0.39\))
If we identify [asymmetry of power \(\leadsto\) asymmetry of outcomes] as harm bc unfair (reflective equilibrium), one “follow-up” policy intervention is make \(A\)’s outside options better (welfare, job retraining, etc… but who sets these up?)
If we identify [ecological damage] as harm, this forms independent “dimension” of policy analysis: if coastal waters are “public good” of Canada, may need some sort of agent representing Canada, to govern use of resource 🤔 some sort of… representative government 🤔 with power to issue fines / ban fishing 🤔

…Fines are “Easy” from Economic Perspective

An economic transaction is a solved political problem. Economics has gained the title “Queen of the Social Sciences” by choosing solved political problems as its domain. (Lerner 1972)

If we assume a “well-functioning” state—power to enforce fines, no corruption, etc.—and that this state has “agreed” to use fines to resolve the issue…
Calculation of “optimal fines” is a “solved” problem in economics (like encryption in CS): A Pigouvian tax just fines agent \(B\) an amount equal to the externality(!) their defection imposes on \(A\), then redistributes this collected fine back to \(A\):

		\(\color{#e69f00}B\)
		Fish 6h		Fish 8h
\(\color{#0072b2}A\)	Fish 6h	\(\color{#0072b2}\cancel{1}\color{black}, \,\)	\(\color{#e69f00}\cancel{1}\)	\(\underset{(+1)}{\color{#0072b2}\cancel{0}\color{black}}\underset{\; \leftarrow\vphantom{(+1)}}{,}\)	\(\underset{(-1)}{\color{#e69f00}\boxed{\color{#e69f00}\mathbf{1.2}}}\)
\(\color{#0072b2}A\)	Fish 8h	\(\underset{(-1)}{\color{#0072b2}\boxed{\color{#0072b2}\mathbf{1.2}}}\color{black}\underset{\rightarrow\vphantom{(+1)}}{,} \,\)	\(\underset{(+1)}{\color{#e69f00}\cancel{0}}\)	\(\color{#0072b2}\boxed{\mathbf{0.4}}\color{black},\)	\(\color{#e69f00}\boxed{\mathbf{0.4}}\)

Figure 1: Original Fishers’ Dilemma, with Pigou fines in parentheses under “agreement-violating” outcomes

\(\leadsto\)
New game with tax applied

		\(\color{#e69f00}B\)
		Fish 6h		Fish 8h
\(\color{#0072b2}A\)	Fish 6h	\(\color{#0072b2}\boxed{\mathbf{1}}\color{black}, \,\)	\(\color{#e69f00}\boxed{\mathbf{1}}\)	\(\color{#0072b2}\boxed{\mathbf{1}}\color{black},\)	\(\color{#e69f00}\cancel{0.2}\)
\(\color{#0072b2}A\)	Fish 8h	\(\color{#0072b2}\cancel{0.2}, \,\)	\(\color{#e69f00}\boxed{\mathbf{1}}\)	\(\color{#0072b2}\cancel{0.4}\color{black},\)	\(\color{#e69f00}\cancel{0.4}\)

Figure 2: The Fishers’ Dilemma with a Pigou tax for unilaterally Fishing 8h

Policy Interventions: Fish Dilemmas \(\leadsto\) Assurance Games

To “escape” prisoners’ dilemma, we had to change the rules of the game (permanently: a one-time fine would not work)
Fishers’ Dilemma:
- No institutions: \(a_A, a_B \in \{6\text{ hr}, 8\text{ hr}\}\)
- Institutions (courts or social norms): \(\{\text{Accept}, \text{Reject}\}\)
Driving “game” (two cars pull up at intersection):
- No institutions: \(a_A, a_B \in \{\text{Stop}, \text{Drive}\}\)
- Institutions (stoplights installed by govt or community agreement): \(a_A, a_B \in \{\text{Obey Light}, \text{Run Light}\}\)
If policy issue well-modeled by Assurance Game, however, may only need to “nudge” (one-time intervention) \(\leadsto\) new permanent Pareto-optimal equilibrium (Nash \(\implies\) self-enforcing!)

Assurance Game

Multiple equilibria; the particular outcome we observe is a function of history (path dependency)
Drive-on-left vs. drive-on-right: Assurance game where neither equilibrium Pareto-dominates other option
- Swedish Dagen H: Nudge from \(o^*_{\textsf{L}} = o(\textsf{L},\textsf{L})\) to \(o^*_{\textsf{R}} = o(\textsf{R},\textsf{R})\)
- Either eq is self-reinforcing! (Unless you… like dying)

QWERTY vs. DVORAK / Palanpur farmers: Assurance game where observed equilibrium Pareto inferior

		\(\color{#e69f00}B\)
		Early	Late
\(\color{#0072b2}A\)	Early	\(\color{#0072b2}\boxed{\mathbf{4}}\color{black}{,} \color{#e69f00}\boxed{\mathbf{4}}\)	\(\color{#0072b2}\cancel{0}\color{black}{,} \color{#e69f00}\cancel{3}\)
\(\color{#0072b2}A\)	Late	\(\color{#0072b2}\cancel{3}\color{black}{,} \color{#e69f00}\cancel{0}\)	\(\color{#0072b2}\boxed{\mathbf{2}}\color{black}{,} \color{#e69f00}\boxed{\mathbf{2}}\)

Invisible Hand Game

Single, unique Nash equilibrium, and it’s Pareto efficient
\(\Rightarrow\) Acting in self interest \(\leadsto\) best possible outcome

It is not from the benevolence of the butcher, the brewer, or the baker that we expect our meal, but from their regard to their own interest (Smith 1776)

		\(\color{#e69f00}B\)
		Corn	Taro
\(\color{#0072b2}A\)	Corn	\(\color{#0072b2}\cancel{2}\color{black}{,} \color{#e69f00}\boxed{\mathbf{4}}\)	\(\color{#0072b2}\boxed{\mathbf{4}}\color{black}{,} \color{#e69f00}\cancel{3}\)
\(\color{#0072b2}A\)	Taro	\(\color{#0072b2}\boxed{\mathbf{5}}\color{black}{,} \color{#e69f00}\boxed{\mathbf{5}}\)	\(\color{#0072b2}\cancel{3}\color{black}{,} \color{#e69f00}\cancel{2}\)

Wealth of Nations SPOILER: The wealth comes from division of labor
and also dumbleydore, and semperus snake, and even poor ron the weasel, who never deserved such a fate

[…I am once again reminding you that] An economic transaction is a solved political problem. Economics gained the title “Queen of the Social Sciences” by choosing solved political problems as its domain

References

Lerner, Abba P. 1972. “The Economics and Politics of Consumer Sovereignty.” The American Economic Review 62 (1/2): 258–66. https://www.jstor.org/stable/1821551.

Smith, Adam. 1776. The Wealth of Nations. Random House Publishing Group. https://books.google.com?id=Anwp7vTR1usC.

Weber, Max. 1919. The Vocation Lectures. Hackett Publishing. https://books.google.com?id=V6t5EAAAQBAJ.

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