Social welfare relations and descriptive set theory

Transcription

1 Albert-Ludwigs-Universität Freiburg AILA meeting - September 2017

2 Infinite utility streams We consider: a set of utility levels Y with some given topology (e.g., Y = {0, 1}, [0, 1], ω) X := Y ω the space of infinite utility streams, endowed with the product topology Given x, y X we use the following notation: x y iff forall n ω, x(n) y(n) x < y iff x y and n ω, x(n) < y(n) F := {π : ω ω : finite permutation} x X, x π := (x π(0), x π(1),..., x π(n),... ). We consider preorders on X.

3 Efficiency conditions Definition Let be a preorder on X (reflexive and transitive relations). is said to be: strongly Paretian iff x < y x y intermediate Paretian iff n(x(n) < y(n)) x y weakly Paretian iff n(x(n) < y(n)) x y.

4 An equity condition Definition A preorder is said to be finitely anonimous iff for every finite permutation π we have x π x.

5 An equity condition Definition A preorder is said to be finitely anonimous iff for every finite permutation π we have x π x. A pre-order which is both finitely anonimous and strong Paretian is called strong ethical preference relations (strong EPR)

6 What about total EPR? Do they exists? Proposition (Folklore) AC implies the existence of total EPR. Proposition (Lawers, 2011) If there is a total EPR, then there is a non-ramsey set. Proposition (Zame, 2007) If there is a total EPR, then there is a non-measurable set.

7 Some easy facts In L there is a 1 2 total EPR. There are no Borel total EPR. an ω 1 -iteration of random forcing kills all 1 2 total EPRs. an ω 1 -iteration of Mathias forcing kills all Σ 1 2 total EPRs.

8 Question: (Bowler, Delhommé, Di Prisco and Mathias, Flutters and Chameleon, Problem 11.14) Is the existence of a total EPR somehow connected with the Baire property?

9 Question: (Bowler, Delhommé, Di Prisco and Mathias, Flutters and Chameleon, Problem 11.14) Is the existence of a total EPR somehow connected with the Baire property? Question: Does the existence of a non-ramsey set imply the existence of a total EPR? Question: Does the existence of a non-measurable set imply the existence of a total EPR?

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11 Idea We first prove that the existence of a total EPR gives a set without Baire property. We then use Shelah s model where all sets have the Baire property (and so there are no total EPR) but there is a non measurable set.

12 Question: How many incompatiple elements are there? We start with a basic example. Let be defined as follows: for every x, y X, we say x y iff there exists π F such that x π < y. Lemma A := {(x, y) X X : x y y x} is comeager.

13 Proof. Let A be the complement of A. We show that A is meager. First note that A can be partitioned into two pieces: E := {(x, y) X X : x y} and D := {(x, y) X X : y x}. We prove E is meager, since the proof for D works similarly. Fix y X so that supp(y) is infinite (i.e., y is not eventually 0) and consider E y := {x X : (x, y) E}. Let H y := {x X : x y}. Note that E y := π F H y π. Since F is countable it is enough to prove that for each π F, H y π is meager.

14 Proof. Actually we show that H y is nowhere dense, for every y X with supp(y) = ω. Indeed, fix U X basic open set, and let k ω be sufficiently large that for all n k, U n = [0, 1]. Then pick n > k such that n supp(y) and pick U U so that: n n, U n = U n; U n := [0, y(n )) Then it is clear that U H y =. This concludes the proof that each H y is nowhere dense, when supp(y) = ω. Note that if π F we get supp(y π) = ω as well, and so H y π is nowhere dense too.

15 Proof. By Ulam-Kuratowski theorem, we conclude the proof if we show that the set {y X : supp(y) = ω} is comeager. So let B be the complement of such a set, i.e., B consists of those y that are eventually 0. Define B n := {y B : supp(y) n}. Clearly B := n ω B n. Moreover each B n in nowhere dense. Indeed, let U be a basic open set and pick k > n so that for all m k, U m = [0, 1]. Then define U U by replacing the kth of U with (0, 1]. It is clear that U B n =. Hence, we have proved that for comeager many y, E y is meager, and that implies E is meager by Kuratowski-Ulam theorem.

16 A generalization of the previous result gives us. Proposition Let be a partial EPR, and A := {(x, y) X X : x y y x}. If A has the Baire property, then A is comeager. Question: But what about total EPR?

17 Proposition Let be a total EPR, and A := {(x, y) X X : x y y x}. Then A does not have the Baire property. Proof. Note that in this case the EPR is total and so the set A = {(x, y) X X : x y}. To reach a contradiction, assume A has the Baire property. By the previous proposition, A has to be comeager.

18 Proof. Hence, by Kuratowski-Ulam s there is y X such that A y is comeager. For 0 < r < 1, define the function i : X X such that i(x(0)) := x(0) + r and n > 0, i(x(n)) = x(n). Note also that for every x X, i(x) x and so in particular x y x i(y). Hence, A y i[a y ] =. Since A y is comeager, it should be A y i[a y ], yielding to a contradiction.

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20 Idea Use Shelah s amalgamation to build a model where all sets in L(R, {Y }) are measurable (and so there are no total EPRs) but Y is non-ramsey. Consider the L(R, {Y }) of such a forcing-extension in order to get a model where all sets are measurable but there is a non-ramsey set.

21 The main property Definition ((B, Y )-homogeneity) Let B be a complete Boolean algebra, Y be B-names. One says that B is (B, Y )-homogeneous if and only if for any isomorphism φ between two complete subalgebras B 1, B 2 of B, such that B 1 B 2 B, there exists φ : B B automorphism extending φ such that B φ ( Y ) = Y. (Intuitively, we want B-names fixed by any automorphism constructed by the amalgamation).

22 Shelah s amalgamation Let B 0, B 1 B isomorphic complete subalgebras and φ : B 0 B 1. Let e 0 : B B B such that e 0 (b) = (b, 1) (and analogously e 1 (b) = (1, b)). Step 1: define Am 1 (B, φ 0 ) B B and φ 1 : e 0 [B] e 1 [B] so that φ 1 is an isomorphism extending φ. Step n: define Am n = Am(Am n 1, φ n 1 ) and φ n extends φ n 1. Step ω: define Am ω (B, φ) as the direct limit of the B n s and φ ω the limit of the φ n s.

23 The main construction Let κ be inaccessible. We recursively build a sequence of complete Boolean algebras {B i : i > κ} and a sequence of sets of names for reals {Y i : i < κ} such that i < j < κ, B i B j and Y i Y j as follows: Using a book-keeping argument we cofinally often amalgamate over random algebras and we fix the set Y i under the isomorphisms generated by the amalgamation. (To get (B, Y )-homogeneity) for cofinally many i we put B i+1 = B i A and Y i+1 = Y i for cofinally many i we put B i+1 = B i MA and Y i+1 = Y i for cofinally many i we put B i+1 = B i MA and Y i+1 = Y i {x T : T MA} at limit steps j < κ put B j = lim i<j B i and Y j = i>j Y i.

24 Two key-steps Dominating reals are preserved under iteration with random forcing. Dominating reals are in a sense preserved by amalgamation. Let B = lim i<κ B i, Y = i<κ Y i and let G be B-generic over V. L(R, {Y }) V [G] = no total EPR and Y is non-ramsey.

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26 Introduction: social welfare relations Egalitarian conditions Compare Paretian principles with the following: Hammond s equity: For every x, y X, if there are i, j N such that x i < y i < y j < x j and for all k i, j one has x k = y k, then x y.

27 Introduction: social welfare relations Egalitarian conditions Compare Paretian principles with the following: Hammond s equity: For every x, y X, if there are i, j N such that x i < y i < y j < x j and for all k i, j one has x k = y k, then x y. Non-dictatorship conditions Investigate non-dictatorship conditions for social choices (Tadeusz Litak, Infinite populations, choice and determinacy, Studia Logica, (2017))

28 Introduction: social welfare relations Egalitarian conditions Compare Paretian principles with the following: Hammond s equity: For every x, y X, if there are i, j N such that x i < y i < y j < x j and for all k i, j one has x k = y k, then x y. Non-dictatorship conditions Investigate non-dictatorship conditions for social choices (Tadeusz Litak, Infinite populations, choice and determinacy, Studia Logica, (2017)) Infinite games Investigate the connections with infinite games.

29 References 1 Haim Judah, Andrej Roslanowsky, On Shelah s amalgamation, Israel Mathematical Conference Proceedings, Vol. 6 (1993), pp Luc Lauwers, Ordering infinite utility streams comes at the cost of a non-ramsey set, Journal of Math. Economics, Volume 46, pp 32-37, (2009). 3 Tadeusz Litak, Infinite populations, choice and determinacy, Studia Logica, (2017). 4 William R. Zame, Can intergenerational equity be operationalized?, Theoretical Economics 2, pp , (2007).

30 Thank you for your attention!