michael brand. game theory = the mathematics of joint decision-making

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The Utilitarian Bargaining Solution Revisited Michael Brand

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The Utilitarian Bargaining Solution

RevisitedMichael Brand

Game Theory=

The mathematics of joint decision-making

Game Theory=

The mathematics of joint decision-making?

Some simple (not-atypical) games

Prisoner’s Dilemma Stag Hunt

Player 1 Player 1

Player 2

(1,1) (0,5) Player 2

(1,1) (0,1)

(5,0) (3,3) (1,0) (2,2)

Battle of the Sexes Chicken

Player 1 Player 1

Player 2

(0,0) (3,2) Player 2

(-9,-9) (-1,1)

(2,3) (0,0) (1,-1) (0,0)

* von Neumann and Morgenstern (1944)

(1,1)

The bargaining problem

(2,3)

(3,2)

(0,0)

SThe utility feasibility set

d

• Convex• Comprehensive• Nontrivial• Closed• Bounded from above

is

The disagreement point

* Nash (1950)

ideal(S)

A bargaining solution

S

d

* Nash (1950)

ψ(S,d) ∈ S ∪ {d}

• feasible

* Nash (1950) + Nash (1953)

Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Invariance to Irrelevant Alternatives (IIA)

The Nash Bargaining Solution

* Nash (1950)

d

iii

dxSx

dx )(maxarg,

d d

INV

WPO

SYM

IIANBS

* Kalai, Smorodinsky (1975)

Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Monotonicity (MONO)

The Kalai-Smorodinsky Solution

d

ideal(S)

A bargaining solution

S

d

* Nash (1950)

ψ(S,d) ∈ S ∪ {d}

• feasible

A bargaining quasi-solution

S

d

ψ(S,d) ⊆ S ∪ {d}

• feasible• nonempty• closed• nontrivial

Social choice

S ψ(S) ⊆ S

• feasible• nonempty• closed• nontrivial

* Harsanyi (1955), Myerson (1981), Thompson (1981)

d

ψ(S,d) ⊆ S ∪ {d}

An explosion of papers

Axioms Solutions

Individual Rationality (IR)

Pareto Optimality (PO) Linearity (LIN) Upper Linearity (ULIN) Concavity (CONC) Individual

Monotonicity (IMONO) etc...

The egalitarian solution

The dictatorial solution

The serial-dictatorial solution

The Yu solutions The Maschler-Perles

solution etc...

The alternating offers game

... and new approaches

d

* Rubinstein (1982)

NBS

Why has nobody heard about any of

this?

Invariance to Irrelevant Alternatives (IIA)◦ How can we know this, until we consider how the

feasibility set is explored? (Kalai+Smorodinsky) Weak Pareto optimality (WPO)

◦ The same question applies.◦ Commodity space may have a different topology.

Symmetry (SYM)◦ Why, exactly, are we assuming this?◦ Is life fair?◦ Are all negotiations symmetric?

Breaking down the old axioms

Life may not be fair, but a good arbitrator should be

Axiom QualityInvariance to Irrelevant Alternatives (IIA)

Thorough

Weak Pareto Optimality (WPO) / Pareto Optimality (PO)

Benevolent

Symmetry (SYM) Impartial

INV claims that by rescaling to other vN-M utility units, the solution cannot be altered.

It is considered to be a statement regarding the inability to compare utility interpersonally.

In fact, it is a stronger statement than this. It is a claim that all arbitrators must necessarily reach the same conclusion, because their decisions must refrain from subjective interpersonal assessment of utilities.

It is a claim that justice is objective.

What about INV?

Does this agree with our intuitive notion of fairness?

d d

INVTo A or to B?

d d

INVStrawberry Shortcake

vs.Lemon Tart

The most we can require of an arbitrator is that her method of interpersonal utility comparisons is consistent.◦ Or else, again, we are back at the “Strawberry

Shortcake vs. Lemon Tart” dilemma. SYM now has to be reformulated.

◦ The arbitrator should now be required to be impartial within her subject world view.

We assume the problem to be scaled into this world view.

So, good arbitration cannot be “objective”

What is the role of d in arbitration? Is the arbitration binding?

◦ If so: no role.◦ If not: shouldn’t S reflect real outcomes, as

opposed to apparent outcomes? This method of modeling actually gives

more modeling power.

Bargaining or social choice?

(4,4,0)

(4,0,4)

(0,4,4) (3,3,3

)

Note: S no longer comprehensive.

What division of the cake should John and Jane decide on, if they are on their way to the shop and still don’t know which cake is in store?

We need a new axiom

LIN

CONC

ex-post efficiency vs. ex-ante efficiency

Myerson (1981)’s “timing effect”

Which brings us to the last quality of a good arbitrator

Axiom QualityInvariance to Irrelevant Alternatives (IIA)

Thorough

Weak Pareto Optimality (WPO) / Pareto Optimality (PO)

Benevolent

Symmetry (SYM) ImpartialConcavity (CONC) Uses foresight

WPO+SYM+IIA+CONC ⇔◦ The Egalitarian Solution or The Utilitarian Quasi-

Solution (for a comprehensive problem domain)

Some of the main results

i

iSx

xSnUtilitaria maxarg)(

i

ixxjiSx

xSnEgalitariaji:,,

maxarg)(

Edgeworth (1881), Walras (1954)

Zeuthen (1930), Harsanyi (1955)

Bentham (1907), Rawls (1971)

Kalai (1977)

“The Veil of Ignorance”

WPO+SYM+IIA+CONC ⇔◦ The Egalitarian Solution or The Utilitarian Quasi-

Solution (for a comprehensive problem domain) But only the Utilitarian Quasi-Solution ⇔

◦ Admitting non-comprehensive problems◦ Strengthening WPO to PO◦ Strengthening CONC to ULIN or to LIN

Some of the main results

One of the tenets of the modern legal system

NBS is a solution, but only when S is guaranteed to be convex.◦ Otherwise, it is a quasi-solution, and is known as

the “Nash Set” The utilitarian quasi-solution is a quasi-

solution on general convex S. However, it is a solution on strictly convex S.

A strictly convex S occurs when goods are infinitely divisible and◦ Players are risk avoiders; or◦ Returns diminish

Solution or quasi-solution?

IIA implies that there is a social utility function

PO implies that this function is monotone increasing in each axis

LIN implies that it is convex SYM implies that it maps all coordinate

permutations to the same value◦ which, together with convexity, leads to being a

function on the sum of the coordinates.

A simplified proof for a simplified case

Shapley (1969)’s “Guiding Principle”:◦ ψ(S) = Efficient(S) ∩ Equitable(S)

Beyond the standard axiomatic model

PO(S) ⊆ Efficient(S) ⊆ WPO(S)

SYM ULINIIA

The Utilitarian Quasi-Solution

* On non-comprehensive domains

Now, we do need to look at the mechanics of haggling.

The mechanics of Rubinstein’s alternating offers game:◦ Infinite turns (or else the solution is dictatorial)◦ Infinite regression of refusals leads to d.◦ Time costs: <Si+1,di+1>=<(1-ε)Si+ε di,di>◦ When ε→0, the first offer is NBS and it is

immediately accepted.

What happens in non-arbitrated scenarios?

<Si+1,di+1>=<(1-ε)Si+ε di,di> Rubinstein: At each offer, there is a 1-ε

probability for negotiations to break down. Why should negotiations ever break down

for rational players? Why at a constant rate? Is it realistic to assume that no amount of

refusals can ever reduce utility to less than a fixed amount?

Is this realistic?

Real-life time costs are exogenous to the bargaining problems

900100800200700300600400Man, I could be at home watching TV right now...

I’d rather be

sailing.

Let A be the vector designating for each player the rate at which her utility is reduced in terms of alternate time costs.

St+Δt={x-AΔt|x ∈ St} We take Δt→0 and tmax→∞. The result is in the utilitarian quasi-solution

Note: The mechanics of the bargaining process dictate the solution’s scaling, with no need for interpersonal utility comparisons.

Real-life time costs are exogenous to the bargaining problems

i

iiSx

Ax /maxarg

W.l.o.g., let us scale the problem to A=1. We know we are on ∂S (the Pareto surface of S), and

because Δt→0 we know S changes slowly. Let p(x)=the normal to ∂S at x (the natural rate of

utility exchange). When backtracking over n turns, the leading offer

changes in direction <1/p1(x)-n/s,... , 1/pn(x)-n/s>, where s=∑pi(x).

Applying the Cauchy-Schwarz inequality, we get that ∑xi always increases, except when p(x)∝1.

Letting tmax→∞, we are guaranteed to reach a point on the utilitarian quasi-solution.

A simplified proof for a simplified case

We know this from experience. We now know that it is rational behavior. It is not accounted for by NBS (Or

Rubinstein’s alternating offers game).

Conclusion: Never haggle when you are in a hurry.

questions?

Thank you!