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Partially speciﬁed probabilities (PSP) Ehud Lehrer Tel Aviv University Tel Aviv, Israel www.tau.ac.il/∼lehrer RUD2007, Tel Aviv Ehud Lehrer – Partially speciﬁed probabilities – p. 1/ A-A (vN-M) Fully Bayesian Partially-specified probabilities (probability of events in sub-algebra) Partially Bayesian Partially-specified probabilities Partially (expectation of random variables) Bayesian Choquet EUM (Schmeidler 1989) Multiple prior (Gilboa-Schmeidler 1989) Ellsberg urn Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black • One ball is randomly drawn from the urn Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black • One ball is randomly drawn from the urn • A decision maker (DM) is asked to choose between two lotteries: Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black • One ball is randomly drawn from the urn • A decision maker (DM) is asked to choose between two lotteries: • X: obtain $100 if Red. Y: obtain $100 if White Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black • One ball is randomly drawn from the urn • A decision maker (DM) is asked to choose between two lotteries: • X: obtain $100 if Red. Y: obtain $100 if White • W: obtain $100 if Red or Black. Z: obtain $100 if White or Black Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn • An urn contains 90 balls of three colors • 30 are Red and the other 60 are either White or Black • One ball is randomly drawn from the urn • A decision maker (DM) is asked to choose between two lotteries: • X: obtain $100 if Red. Y: obtain $100 if White • W: obtain $100 if Red or Black. Z: obtain $100 if White or Black • If you prefer X over Y and Z over W you are not an expected utility maximizer Ehud Lehrer – Partially speciﬁed probabilities – p. 1/? Ellsberg urn - cont. Ehud Lehrer – Partially speciﬁed probabilities – p. 2/? Ellsberg urn - cont. • The DM ought to take decisions while having only partial information about the real distribution Ehud Lehrer – Partially speciﬁed probabilities – p. 2/? Ellsberg urn - cont. • The DM ought to take decisions while having only partial information about the real distribution • From the information DM obtained she can infer that P(Red) = 1/3 and I I P(White or Black) = 2/3. Ehud Lehrer – Partially speciﬁed probabilities – p. 2/? Dynamic urn Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn • Suppose that on day 2 the number of White balls is doubled Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn • Suppose that on day 2 the number of White balls is doubled • What is now the distribution of colors? Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn • Suppose that on day 2 the number of White balls is doubled • What is now the distribution of colors? • The probability of no non-trivial event is known: the prob. of Red is no longer 1/3; that of White or Black is no longer 2/3 Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn • Suppose that on day 2 the number of White balls is doubled • What is now the distribution of colors? • The probability of no non-trivial event is known: the prob. of Red is no longer 1/3; that of White or Black is no longer 2/3 • So, after all what the DM knows? Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn • Suppose that on day 2 the number of White balls is doubled • What is now the distribution of colors? • The probability of no non-trivial event is known: the prob. of Red is no longer 1/3; that of White or Black is no longer 2/3 • So, after all what the DM knows? • DM knows the expectation of some, but not all, random variables Ehud Lehrer – Partially speciﬁed probabilities – p. 3/? Dynamic urn - cont. Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 • Denote by ni the number of color i balls at day one Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 • Denote by ni the number of color i balls at day one R W B (nW + nB = 60) nR = 30 nW nB Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 • Denote by ni the number of color i balls at day one R W B (nW + nB = 60) nR = 30 nW nB 1 nR n 3 W 1 nR +nW +nB = nW = 3 Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 • Denote by ni the number of color i balls at day one R W B (nW + nB = 60) nR = 30 nW nB 1 nR n 3 W 1 nR +nW +nB = nW = 3 • On day 2 there are 2nW White balls Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - cont. 1 • Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)] 1 E(X) = • We’ll see that I 3 • Denote by ni the number of color i balls at day one R W B (nW + nB = 60) nR = 30 nW nB 1 nR n 3 W 1 nR +nW +nB = nW = 3 • On day 2 there are 2nW White balls 1nR + 1 2nW +0nB 1 nR + 3 nW 1 E(X) = I 6 nR +2nW +nB = nR +nW +nB +nW = 3 Ehud Lehrer – Partially speciﬁed probabilities – p. 4/? Dynamic urn - conclusion Ehud Lehrer – Partially speciﬁed probabilities – p. 5/? Dynamic urn - conclusion • On day 2 the expectation of two random variables is known: Ehud Lehrer – Partially speciﬁed probabilities – p. 5/? Dynamic urn - conclusion • On day 2 the expectation of two random variables is known: • X = [1(R); 6 (W ); 0(B)] (expectation= 1 ) and 1 3 [1(R); 1(W ); 1(B)] (expectation=1). Ehud Lehrer – Partially speciﬁed probabilities – p. 5/? Dynamic urn - conclusion • On day 2 the expectation of two random variables is known: • X = [1(R); 6 (W ); 0(B)] (expectation= 1 ) and 1 3 [1(R); 1(W ); 1(B)] (expectation=1). • The expectation of any r.v. in the algebra they generate is also known. Ehud Lehrer – Partially speciﬁed probabilities – p. 5/? Noisy signals Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: x y z [1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)] 2 2 Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: x y z [1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)] 2 2 • Suppose it is repeated many times and dist. over x, y and z is uniforms and i.i.d. Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: x y z [1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)] 2 2 • Suppose it is repeated many times and dist. over x, y and z is uniforms and i.i.d. • DM observes 50% the payoff 5 and 50% the payoff 7 Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: x y z [1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)] 2 2 • Suppose it is repeated many times and dist. over x, y and z is uniforms and i.i.d. • DM observes 50% the payoff 5 and 50% the payoff 7 • Asymptotically, all DM knows is that the expectation of [1(x), 0(y), -1(z)] is 0. Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Noisy signals • There are three states of nature: x, y and z • There are two payoffs, 5 and 7 – stochastically depend on the state: x y z [1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)] 2 2 • Suppose it is repeated many times and dist. over x, y and z is uniforms and i.i.d. • DM observes 50% the payoff 5 and 50% the payoff 7 • Asymptotically, all DM knows is that the expectation of [1(x), 0(y), -1(z)] is 0. • In an interactive model when players get only noisy signals about others, players face partially-speciﬁed probability (PSP) Ehud Lehrer – Partially speciﬁed probabilities – p. 6/? Example by Machina (2007) Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. • A decision maker (DM) chooses an act, then a ball is randomly drawn and a reward (utility) is given. The following table summarizes the rewards related to four acts. Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. • A decision maker (DM) chooses an act, then a ball is randomly drawn and a reward (utility) is given. The following table summarizes the rewards related to four acts. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. • A decision maker (DM) chooses an act, then a ball is randomly drawn and a reward (utility) is given. The following table summarizes the rewards related to four acts. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • Notice: f3 is a mirror image of f2 and f4 of f1 Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. • A decision maker (DM) chooses an act, then a ball is randomly drawn and a reward (utility) is given. The following table summarizes the rewards related to four acts. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • We expect (as long as symmetry is kept) that if f2 f1 then f3 f4 Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) • An urn contains 20 balls of four different colors a,b,c, and d. 10 are either a or b and the other 10 are either c or d. • A decision maker (DM) chooses an act, then a ball is randomly drawn and a reward (utility) is given. The following table summarizes the rewards related to four acts. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • We expect (as long as symmetry is kept) that if f2 f1 then f3 f4 , but this is inconsistent with Choquet EUM Ehud Lehrer – Partially speciﬁed probabilities – p. 7/? Example by Machina (2007) - cont. Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 1 P(a, b) = I • It is known that I P(c, d) = 2 Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • It is known that I P(c, d) = 1 P(a, b) = I 2 1 E(X1 ) = • Equivalently, X1 = [1, 1, 0, 0], I 2 and 1 E(X2 ) = 2 X2 = [0, 0, 1, 1], I Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • It is known that I P(c, d) = 1 P(a, b) = I 2 1 E(X1 ) = • Equivalently, X1 = [1, 1, 0, 0], I 2 and 1 E(X2 ) = 2 X2 = [0, 0, 1, 1], I • Belief: P(b, c) is positive. Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • It is known that I P(c, d) = 1 P(a, b) = I 2 1 • Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and 1 E(X2 ) = 2 X2 = [0, 0, 1, 1], I 1 • Belief: P(b, c) is positive. It implies IE([0, 1, 1, 0]) ≥ 20 Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • It is known that I P(c, d) = 1 P(a, b) = I 2 1 • Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and 1 E(X2 ) = 2 X2 = [0, 0, 1, 1], I 1 • Belief: P(b, c) is positive. It implies I E([0, 1, 1, 0]) ≥ 20 1 • Let’s assume that I E([0, 1, 1, 0]) = 20 Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Example by Machina (2007) - cont. a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • It is known that I P(c, d) = 1 P(a, b) = I 2 1 • Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and 1 E(X2 ) = 2 X2 = [0, 0, 1, 1], I 1 • Belief: P(b, c) is positive. It implies I E([0, 1, 1, 0]) ≥ 20 1 • Let’s assume that I E([0, 1, 1, 0]) = 20 • Now the expectations of X1 , X2 and [0, 1, 1, 0] are known. Ehud Lehrer – Partially speciﬁed probabilities – p. 8/? Partially-speciﬁed probability (PSP) Ehud Lehrer – Partially speciﬁed probabilities – p. 9/? Partially-speciﬁed probability (PSP) P, • Partially-speciﬁed probability is a pair (I Y) Ehud Lehrer – Partially speciﬁed probabilities – p. 9/? Partially-speciﬁed probability (PSP) P, • Partially-speciﬁed probability is a pair (I Y) P • I – probability Ehud Lehrer – Partially speciﬁed probabilities – p. 9/? Partially-speciﬁed probability (PSP) P, • Partially-speciﬁed probability is a pair (I Y) P • I – probability • Y – a set of random variables Ehud Lehrer – Partially speciﬁed probabilities – p. 9/? Partially-speciﬁed probability (PSP) P, • Partially-speciﬁed probability is a pair (I Y) P • I – probability • Y – a set of random variables EP E(Y ), Y ∈ Y • DM is informed only of I I (Y ) = I Ehud Lehrer – Partially speciﬁed probabilities – p. 9/? How one can use PSP? Ehud Lehrer – Partially speciﬁed probabilities – p. 10/? How one can use PSP? P, • Let (I Y) be PSP (partially-speciﬁed probability) Ehud Lehrer – Partially speciﬁed probabilities – p. 10/? How one can use PSP? P, • Let (I Y) be PSP (partially-speciﬁed probability) • X is a random variable Ehud Lehrer – Partially speciﬁed probabilities – p. 10/? How one can use PSP? P, • Let (I Y) be PSP (partially-speciﬁed probability) • X is a random variable P, • Deﬁne the expectation of X w.r.t (I Y), Ehud Lehrer – Partially speciﬁed probabilities – p. 10/? How one can use PSP? P, • Let (I Y) be PSP (partially-speciﬁed probability) • X is a random variable P, • Deﬁne the expectation of X w.r.t (I Y), EP I (I ,Y) (X) = max{ E(Y ); αY I αY Y ≤ X and αY ∈ R} Ehud Lehrer – Partially speciﬁed probabilities – p. 10/? Evaluating acts with PSP Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Evaluating acts with PSP P, • Suppose that (I Y) is a PSP over S (the state space) Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Evaluating acts with PSP P, • Suppose that (I Y) is a PSP over S (the state space) • and u is an afﬁne (utility function) deﬁnes over ∆(L) Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Evaluating acts with PSP P, • Suppose that (I Y) is a PSP over S (the state space) • and u is an afﬁne (utility function) deﬁnes over ∆(L) • Note that if f is an act, then u ◦ f is a random variable deﬁned over S Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Evaluating acts with PSP P, • Suppose that (I Y) is a PSP over S (the state space) • and u is an afﬁne (utility function) deﬁnes over ∆(L) • Note that if f is an act, then u ◦ f is a random variable deﬁned over S • PSP and u induce a complete order over acts. Let f and g be acts, then Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Evaluating acts with PSP P, • Suppose that (I Y) is a PSP over S (the state space) • and u is an afﬁne (utility function) deﬁnes over ∆(L) • Note that if f is an act, then u ◦ f is a random variable deﬁned over S • PSP and u induce a complete order over acts. Let f and g be acts, then f EP EP g iﬀ I (I ,Y) (u ◦ f ) ≥ I (I ,Y) (u ◦ g) Ehud Lehrer – Partially speciﬁed probabilities – p. 11/? Back to Machina’s Example Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 1 1 • Recall, IE(X1 ) = I E(X3 ) = E(X2 ) = 2 and I 20 P, (X3 = [0, 1, 1, 0]). This deﬁned PSP (I Y) Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • f1 corresponds to Y = [0, 200, 100, 100] and f2 to Z = [0, 100, 200, 100] (u identity – the reward in utile terms) Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • f1 corresponds to Y = [0, 200, 100, 100] and f2 to Z = [0, 100, 200, 100] (u identity – the reward in utile terms) • Z = [0, 100, 100, 0] + [0, 0, 100, 100] and I E(Z) = 100I E(X3 ) + 100I X2 = 100 + 100 E 20 2 Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • f1 corresponds to Y = [0, 200, 100, 100] and f2 to Z = [0, 100, 200, 100] (u identity – the reward in utile terms) • Z = [0, 100, 100, 0] + [0, 0, 100, 100] and I E(Z) = 100I E(X3 ) + 100I X2 = 100 + 100 E 20 2 • The best approximation from below to Y with linear combinations of X1 , X2 and X3 is 100X2 Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • f1 corresponds to Y = [0, 200, 100, 100] and f2 to Z = [0, 100, 200, 100] (u identity – the reward in utile terms) • Z = [0, 100, 100, 0] + [0, 0, 100, 100] and I E(Z) = 100I E(X3 ) + 100I X2 = 100 + 100 E 20 2 • The best approximation from below to Y with linear combinations of X1 , X2 and X3 is 100X2 EP EP • Thus, I (I ,Y) (Z) > I (I ,Y) (Y ), which implies f2 f1 Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Back to Machina’s Example a b c d f1 0 200 100 100 f2 0 100 200 100 f3 100 200 100 0 f4 100 100 200 0 • f1 corresponds to Y = [0, 200, 100, 100] and f2 to Z = [0, 100, 200, 100] (u identity – the reward in utile terms) • Z = [0, 100, 100, 0] + [0, 0, 100, 100] and I E(Z) = 100I E(X3 ) + 100I X2 = 100 + 100 E 20 2 • The best approximation from below to Y with linear combinations of X1 , X2 and X3 is 100X2 EP EP • Thus, I (I ,Y) (Z) > I (I ,Y) (Y ), which implies f2 f1 • From similar reasons, f3 f4 Ehud Lehrer – Partially speciﬁed probabilities – p. 12/? Axiomatization Ehud Lehrer – Partially speciﬁed probabilities – p. 13/? Axiomatization • Pretty much like Anscombe-Aumann with a different version of the independence axiom Ehud Lehrer – Partially speciﬁed probabilities – p. 13/? Axiomatization • Pretty much like Anscombe-Aumann with a different version of the independence axiom • Deﬁnition: An act f is fat free (FaF) if f (s) g(s) for every s ∈ S with at least one strict preference, implies f g Ehud Lehrer – Partially speciﬁed probabilities – p. 13/? Axiomatization • Pretty much like Anscombe-Aumann with a different version of the independence axiom • Deﬁnition: An act f is fat free (FaF) if f (s) g(s) for every s ∈ S with at least one strict preference, implies f g • Deﬁnition: An act f is strongly fat free (SFaF) if for every constant act c and every α ∈ (0, 1), αf + (1 − α)c is fat free Ehud Lehrer – Partially speciﬁed probabilities – p. 13/? Axiomatization • Pretty much like Anscombe-Aumann with a different version of the independence axiom • Deﬁnition: An act f is fat free (FaF) if f (s) g(s) for every s ∈ S with at least one strict preference, implies f g • Deﬁnition: An act f is strongly fat free (SFaF) if for every constant act c and every α ∈ (0, 1), αf + (1 − α)c is fat free • SFaF independence: if f and g are acts such that f g and h is SFaF, then αf + (1 − α)h αg + (1 − α)h for every α ∈ (0, 1] Ehud Lehrer – Partially speciﬁed probabilities – p. 13/? Relation to Choquet EUM (Schmeidler 1989) Ehud Lehrer – Partially speciﬁed probabilities – p. 14/? Relation to Choquet EUM (Schmeidler 1989) • If in PSP Y consists of the indicators of events in a sub-algebra, then a convex capacity can be deﬁned. Ehud Lehrer – Partially speciﬁed probabilities – p. 14/? Relation to Choquet EUM (Schmeidler 1989) • If in PSP Y consists of the indicators of events in a sub-algebra, then a convex capacity can be deﬁned. • On convex capacities the concave integral (Lehrer, 2005) coincides with Choquet integral • Thus, when Y consists of the indicators of events in a sub-algebra the PSP model is strictly between A-A and Choquet EUM Ehud Lehrer – Partially speciﬁed probabilities – p. 14/? Relation to Choquet EUM (Schmeidler 1989) • If in PSP Y consists of the indicators of events in a sub-algebra, then a convex capacity can be deﬁned. • On convex capacities the concave integral (Lehrer, 2005) coincides with Choquet integral • Thus, when Y consists of the indicators of events in a sub-algebra the PSP model is strictly between A-A and Choquet EUM Ehud Lehrer – Partially speciﬁed probabilities – p. 14/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y P, • Thus, (I Y) induces a set of probability distributions that are consistent with the information available. DM minimizes over these distributions Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y P, • Thus, (I Y) induces a set of probability distributions that are consistent with the information available. DM minimizes over these distributions • This set is an afﬁne space of distributions (intersection of an afﬁne space with the simplex of distributions) Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y P, • Thus, (I Y) induces a set of probability distributions that are consistent with the information available. DM minimizes over these distributions • This set is an afﬁne space of distributions (intersection of an afﬁne space with the simplex of distributions) • The PSP model is strictly between A-A and the Multiple Prior model Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y P, • Thus, (I Y) induces a set of probability distributions that are consistent with the information available. DM minimizes over these distributions • This set is an afﬁne space of distributions (intersection of an afﬁne space with the simplex of distributions) • The PSP model is strictly between A-A and the Multiple Prior model • The PSP model is information-based: the set of priors is determined by the information structure and the real distribution Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989) • From duality: estimating the expectation of a random variable by members of Y is equivalent to minimizing wrt to P probability distributions that agree with I on Y P, • Thus, (I Y) induces a set of probability distributions that are consistent with the information available. DM minimizes over these distributions • This set is an afﬁne space of distributions (intersection of an afﬁne space with the simplex of distributions) • The PSP model is strictly between A-A and the Multiple Prior model • The PSP model is information-based: the set of priors is determined by the information structure and the real distribution • This fact is essential when considering interactive models. Players play actual strategies. The beliefs are determined by the information player get about them Ehud Lehrer – Partially speciﬁed probabilities – p. 15/? Non-cooperative games Ehud Lehrer – Partially speciﬁed probabilities – p. 16/? Non-cooperative games • A game consists of: Ehud Lehrer – Partially speciﬁed probabilities – p. 16/? Non-cooperative games • A game consists of: • N – a ﬁnite set of players Ehud Lehrer – Partially speciﬁed probabilities – p. 16/? Non-cooperative games • A game consists of: • N – a ﬁnite set of players • Ai – A ﬁnite set of player i’s actions Ehud Lehrer – Partially speciﬁed probabilities – p. 16/? Non-cooperative games • A game consists of: • N – a ﬁnite set of players • Ai – A ﬁnite set of player i’s actions • ui – player i’s utility function. R ui : ×Ai → I Ehud Lehrer – Partially speciﬁed probabilities – p. 16/? Partially-speciﬁed equilibrium Ehud Lehrer – Partially speciﬁed probabilities – p. 17/? Partially-speciﬁed equilibrium • Each player plays a pure or a mixed strategy Ehud Lehrer – Partially speciﬁed probabilities – p. 17/? Partially-speciﬁed equilibrium • Each player plays a pure or a mixed strategy • Each player obtains partial information about other players’ strategies Ehud Lehrer – Partially speciﬁed probabilities – p. 17/? Partially-speciﬁed equilibrium • Each player plays a pure or a mixed strategy • Each player obtains partial information about other players’ strategies • Each player maximizes her payoff against the worst (uncoordinated/independent) strategy consistent with her information Ehud Lehrer – Partially speciﬁed probabilities – p. 17/? Example Ehud Lehrer – Partially speciﬁed probabilities – p. 18/? Example • Consider the coordination game L M R T 3,3 0,0 0,0 C 0,0 2,2 0,0 B 0,0 0,0 1,1 Ehud Lehrer – Partially speciﬁed probabilities – p. 18/? Example • Consider the coordination game L M R T 3,3 0,0 0,0 C 0,0 2,2 0,0 B 0,0 0,0 1,1 3 3 • There are three equilibria: a. p = ( 2 , 5 , 0), q = ( 2 , 5 , 0); b. 5 5 2 3 6 p = (0, 0, 1), q = (0, 0, 1); and c. p = ( 11 , 11 , 11 ), 2 3 6 q = ( 11 , 11 , 11 ). Ehud Lehrer – Partially speciﬁed probabilities – p. 18/? Best response Ehud Lehrer – Partially speciﬁed probabilities – p. 19/? Best response • Suppose that player i is informed only of the variables over A−i in Yi Ehud Lehrer – Partially speciﬁed probabilities – p. 19/? Best response • Suppose that player i is informed only of the variables over A−i in Yi • p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes I (p2 ,Y1 ) ( payoff of player 1) E Ehud Lehrer – Partially speciﬁed probabilities – p. 19/? Best response • Suppose that player i is informed only of the variables over A−i in Yi • p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes I (p2 ,Y1 ) ( payoff of player 1) E • Deﬁnition: (p1 , p2 ) is partially speciﬁed probability w.r.t. to the information Y1 and Y2 if pi is a best response to p−i (w.r.t Yi ) Ehud Lehrer – Partially speciﬁed probabilities – p. 19/? Best response • Suppose that player i is informed only of the variables over A−i in Yi • p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes I (p2 ,Y1 ) ( payoff of player 1) E • Deﬁnition: (p1 , p2 ) is partially speciﬁed probability w.r.t. to the information Y1 and Y2 if pi is a best response to p−i (w.r.t Yi ) • In n-player games: pi is a best response to the independent p−i (w.r.t Yi ) Ehud Lehrer – Partially speciﬁed probabilities – p. 19/? Information-based Ehud Lehrer – Partially speciﬁed probabilities – p. 20/? Information-based • The notion of partially-speciﬁed equilibrium is information-based Ehud Lehrer – Partially speciﬁed probabilities – p. 20/? Information-based • The notion of partially-speciﬁed equilibrium is information-based • The information structure, namely, the information available to each player, determines the set of equilibria Ehud Lehrer – Partially speciﬁed probabilities – p. 20/? Information-based • The notion of partially-speciﬁed equilibrium is information-based • The information structure, namely, the information available to each player, determines the set of equilibria • No player has a prior belief about other players’ strategies. The information structure is exogenous Ehud Lehrer – Partially speciﬁed probabilities – p. 20/? Information-based • The notion of partially-speciﬁed equilibrium is information-based • The information structure, namely, the information available to each player, determines the set of equilibria • No player has a prior belief about other players’ strategies. The information structure is exogenous • The belief a player has about other players is determined by the actual strategies, as well as by the partial information a player obtains about them Ehud Lehrer – Partially speciﬁed probabilities – p. 20/? PS correlated equilibrium (PSCE) Ehud Lehrer – Partially speciﬁed probabilities – p. 21/? PS correlated equilibrium (PSCE) • A mediator select (p1 , ..., pn ) according to distribution Q Ehud Lehrer – Partially speciﬁed probabilities – p. 21/? PS correlated equilibrium (PSCE) • A mediator select (p1 , ..., pn ) according to distribution Q • Players don’t know Q Ehud Lehrer – Partially speciﬁed probabilities – p. 21/? PS correlated equilibrium (PSCE) • A mediator select (p1 , ..., pn ) according to distribution Q • Players don’t know Q • The player gets a recommendation to play pi and a partially-speciﬁed probability about Q(p−i |pi ) Ehud Lehrer – Partially speciﬁed probabilities – p. 21/? PS correlated equilibrium (PSCE) • A mediator select (p1 , ..., pn ) according to distribution Q • Players don’t know Q • The player gets a recommendation to play pi and a partially-speciﬁed probability about Q(p−i |pi ) • And he plays a best response Ehud Lehrer – Partially speciﬁed probabilities – p. 21/? Learning to play PSCE • Recall the example with the noisy signals Ehud Lehrer – Partially speciﬁed probabilities – p. 22/? Learning to play PSCE • The game is played repeatedly Ehud Lehrer – Partially speciﬁed probabilities – p. 22/? Learning to play PSCE • The game is played repeatedly • Players observe noisy signals of past actions Ehud Lehrer – Partially speciﬁed probabilities – p. 22/? Learning to play PSCE • The game is played repeatedly • Players observe noisy signals of past actions • Each player plays a conditional regret-free strategy (“I have no regret for playing p because this is the best I could do in response to the worst strategy of the others that is consistent with the signals I received during the times I was playing p.") Ehud Lehrer – Partially speciﬁed probabilities – p. 22/? Learning to play PSCE • The game is played repeatedly • Players observe noisy signals of past actions • Each player plays a conditional regret-free strategy (“I have no regret for playing p because this is the best I could do in response to the worst strategy of the others that is consistent with the signals I received during the times I was playing p.") •A recent paper with Eilon Solan ( “Learning to play partially-speciﬁed equilibrium") shows that conditional regret-free strategies exist and the empirical frequency of the mixed strategies played converges to PSCE Ehud Lehrer – Partially speciﬁed probabilities – p. 22/? A-A (vN-M) Fully Bayesian Partially-specified probabilities (probability of events in sub-algebra) Partially Bayesian Partially-specified probabilities Partially (expectation of random variables) Bayesian Choquet EUM (Schmeidler 1989) Multiple prior (Gilboa-Schmeidler 1989)

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Ehud Lehrer, Tel Aviv, Statistics Department, Visiting Professor, Game Theory, social profiles, random variable, probability distributions, Noisy signals, Tel Aviv University

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posted: | 6/24/2011 |

language: | English |

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