The Value of Information Under Unawareness

The Under Unawareness Spyros Galanis University of Southampton January 30, 2014 The Under Unawareness

Environment Agents invest in the stock market Their payoff is determined by their investments and the prices of the shares Each agent is aware of his payoffs and actions The Under Unawareness

Environment Agents invest in the stock market Their payoff is determined by their investments and the prices of the shares Each agent is aware of his payoffs and actions But he may be unaware of other contingencies like Possibility of a merger The characteristics of a CEO An upcoming innovation The Under Unawareness

Contribution of the paper We examine the value of information, in the presence of unawareness, in the following two settings A single agent who receives a signal and takes an action A competitive risk sharing environment with many agents We show with examples that unawareness can make the value of information negative in the single-agent case and positive in the multi-agent setting, which is opposite of what is true when there is no unawareness The Under Unawareness

Contribution of the paper The reason is that agents misunderstand their awareness signal We identify different patterns of awareness and show which are sufficient for the value of information to be positive in the single-agent and negative in the multi-agent setting The conditions for the multi-agent case are stronger than the conditions for the single-agent case The Under Unawareness

Related Literature 1. Unawareness Fagin and Halpern (1988), Dekel, Lipman and Rustichini (1998), Heifetz, Meier and Schipper (2007), Li (2009), Galanis (2013) 2. - Single Agent Blackwell (1951), Geanakoplos (1989), Schlee (1991), Morris (1992), Morris and Shin (1997), Quiggin (2013) 3. - Hirshleifer (1971), Green (1981), Campbell (2004), Schlee (2001) The Under Unawareness

Figure: The Under Unawareness State Space Single-agent environment

Figure: The Under Unawareness Coarse Partition Single-agent environment

Figure: The Under Unawareness Finer Partition Single-agent environment

Single-agent environment Positive value of information without unawareness More information is represented by a finer partition More flexibility in choosing actions Suppose same prior, utilities and action sets for both agents. Then, the more informed agent has always (weakly) higher ex ante utility The Under Unawareness

Example with Unawareness Single-agent environment Three dimensions: p: What is the price? (l, m, h) q: Is there an acquisition? (y, n) r: Is there an innovation? (y, n) Payoffs depend on prices p l p m p h B -1-1 1 NB 0 0 0 The Under Unawareness

Example Single-agent environment Full state space S contains three states: s 1 = (p l, q y, r n ), s 2 = (p m, q n, r y ), s 3 = (p h, q y, r y ) The Under Unawareness

Single-agent environment Example Full state space S contains three states: s 1 = (p l, q y, r n ), s 2 = (p m, q n, r y ), s 3 = (p h, q y, r y ) The agent is aware of p, q always He is aware of r only at s3 Hence, at s1, s 2, he is aware of state space S 0 s 1 = (p l, q y ), s 2 = (p m, q n ), s 3 = (p h, q y ) The Under Unawareness

Figure: The Under Unawareness Two state spaces - Awareness signal Single-agent environment s* 1 =(p l, q y, r n ) s* 2 =(p m, q n, r y ) s* 3 =(p h, q y, r y ) s 1 =(p l, q y ) s 2 =(p m, q n ) s 3 =(p h, q y )

Single-agent environment Information Signal Suppose there are two agents, identical in everything except information: Agent 2 always knows the answer to question q The Under Unawareness

Figure: The Under Unawareness Information signal Single-agent environment s* 1 =(p l, q y, r n ) s* 2 =(p m, q n, r y ) s* 3 =(p h, q y, r y ) s 1 =(p l, q y ) s 2 =(p m, q n ) s 3 =(p h, q y )

Figure: The Under Unawareness Information and awareness signals Single-agent environment s* 1 =(p l, q y, r n ) s* 2 =(p m, q n, r y ) s* 3 =(p h, q y, r y ) s 1 =(p l, q y ) s 2 =(p m, q n ) s 3 =(p h, q y )

Single-agent environment Optimal Actions The agents have a common prior of (0.3, 0.3, 0.4) Posteriors and optimal actions are as follows Full state Agent 1 Action Agent 2 Action s1 (0.3, 0.3, 0.4) NB (3/7, 0, 4/7) B s2 (0.3, 0.3, 0.4) NB (0, 1, 0) NB s3 (0, 0, 1) B (0, 0, 1) B The more informed agent 2 chooses the wrong action at p l What goes wrong? Awareness signal is used asymmetrically When receiving the high awareness signal, he excludes states that describe a low awareness signal, but the reverse is not true The Under Unawareness

Single-agent environment Negative value of information in standard model Suppose agents trade a single good after receiving a public signal Schlee (2001) showed that the value of information is negative for everyone if 1. All agents are risk averse and there is no aggregate uncertainty 2. The risk averse agents are fully insured by a risk neutral agent The Under Unawareness

Single-agent environment Environment 1 physical good, no aggregate uncertainty 2 risk averse agents with utility u(c) = c 1/4c 2 Agents receive the same public signal, so information and awareness is symmetric Endowments are defined on the least expressive state space S 0 State Agent 1 Agent 2 s 1 1 0 s 2 0.5 0.5 s 3 0 1 The Under Unawareness

Single-agent environment No information The priors are, as before, (0.3, 0.3, 0.4) If there is no information, the risk averse agents want full insurance and demand their expected endowment Agent c(s 1 ) c(s 2 ) c(s 3 ) Exp U 1 0.45 0.45 0.45 0.39 2 0.55 0.55 0.55 0.47 The Under Unawareness

Single-agent environment More information Suppose information is identical to that of agent 2 in previous example At s1, posteriors are (3/7, 0, 4/7) and the allocation is State Agent 1 Agent 2 s 1 3/7 4/7 s 2 0 0 s 3 3/7 4/7 The Under Unawareness

Single-agent environment More information At s2, posteriors are (0, 1, 0) and both agents consume their endowments, which is 0.5 At s3, posteriors are (0, 0, 1) and both agents consume their endowments, which is 0 for 1 and 1 for 2 From an ex ante perspective, information makes 1 strictly worse off and 2 strictly better off Agent No information Information 1 0.39 0.24 2 0.47 0.57 The Under Unawareness

Preliminaries Signals Properties of unawareness The Let S = {S a } a A be a finite collection of disjoint state spaces S is a lattice, endowed with a partial order : S S means S is more expressive than S The collection Σ = a A S a of all states is finite The most complete state space is S The least complete, payoff relevant, state space is S 0 S i (s) denotes i s state space at s The Under Unawareness

Preliminaries Signals Properties of unawareness Projections and enlargements of events An event E is a subset of a state space S Suppose S S S and E S We require a surjective projection rs S : S S E S = {rs S (s ) S : s E } is the projection of E to the less expressive state space S E S = {s S : r S S (s ) E } is the enlargement of E to the less expressive state space S The Under Unawareness

Possibility correspondence Preliminaries Signals Properties of unawareness P i : Σ 2 Σ \ describes i s information signal (0) Confinedness: If s S then P i (s) S for some S S (1) Generalized Reflexivity: s (P i (s)) for every s Σ (2) Stationarity: s P i (s) implies P i (s ) = P i (s) (3) Projections Preserve Ignorance: If s S and S S then (P i (s)) (P i (s S )) (4) Projections Preserve Awareness: If s S, s P i (s) and S S then s S P i (s S ) The Under Unawareness

Generalized Prior Preliminaries Signals Properties of unawareness Let π : Σ [0, 1] be a function such that π(s ) = 1 s S and, for each s Σ, π(s) = π(s ) s s S We call π a generalized prior The Under Unawareness

Information signal Preliminaries Signals Properties of unawareness P 2 is more informative than P 1 given S S, if, for all s S, P 2 (s) S P 1 (s) S The Under Unawareness

Awareness signal Preliminaries Signals Properties of unawareness E i S : S 2S \ describes i s awareness signal, given state space S: E i S (s) = {s 1 S : S i (s) = S i (s 1 )} partitions S, where each partition cell specifies the same awareness for i E i S Definition P i is more informative than the awareness signal of P j given S S if P i (s) S E j S (s) for all s S The Under Unawareness

Conditional Independence Preliminaries Signals Properties of unawareness Definition (P, π) satisfies Conditional Independence given S S if, for any s S with π(s) > 0, for any E S i (s), π(e P i (s)) = π(e E i S (s) Pi (s) S ) Posterior beliefs (conditioning on i s information signal) do not change when conditioning also on i s awareness signal The Under Unawareness

Path-independence Preliminaries Signals Properties of unawareness Definition Given S S, awareness for P i is path-independent if, for all s 1, s 2, s 3 S, if S i (s 3 ) S i (s 1 ) and S i (s 3 ) S i (s 2 ) then S i (s 1 ) S i (s 2 ) or S i (s 2 ) S i (s 1 ) Each level of awareness is reached by at most one (maximal) path of successively more expressive state spaces The Under Unawareness

Single-agent Comparison with HMS Environment (S, P i, C, u, π) is a decision problem where, Uncertainty is described in state space S P i is the agent s possibility correspondence C is the action set u : C S R is his utility function π is a generalized prior on Σ Payoffs depend on the least expressive state space S 0 = S For all s Σ, u(c, s S0 ) = u(c, s) for all c C. The Under Unawareness

Decision functions Single-agent Comparison with HMS Decision function given S, f : S C Let F S be the collection of decision functions f F S is measurable with respect to P i if, for all s 1, s 2 S, P i (s 1 ) = P i (s 2 ) = f (s 1 ) = f (s 2 ) f is optimal for (S, P i, C, u, π) if it is measurable with respect to P i and, for all s S and c C, u(f (s), s )π(s ) u(c, s )π(s ) s P i (s) s P i (s) The Under Unawareness

Single-agent Comparison with HMS Decision problem A = (S, P 1, C, u, π) is more valuable than B = (S, P 2, C, u, π) if, whenever g is optimal for A and f is optimal for B, we have s)π(s) s Su(g(s), u(f (s), s)π(s) s S The Under Unawareness

Single-agent Comparison with HMS Theorem 1 Suppose that P 2 is more informative than P 1 given S S. Decision problem (S, P 2, C, u, π) is more valuable than (S, P 1, C, u, π) if one of the following is true: 1. (P 2, π) satisfies Conditional Independence, or, 2. Awareness for P 2 is path-independent and P 2 is more informative than the awareness signal of P 1, given S The Under Unawareness

Single-agent Comparison with HMS Environment (S, P, ω, u, π) is an exchange economy with I agents where, Uncertainty is described in state space S Information and awareness are common, given by P There is no aggregate risk if the aggregate endowment ω s is constant for all states s S Each agent has a strictly increasing and concave u i Common prior π The Under Unawareness

Definition of equilibrium Single-agent Comparison with HMS Given an economy (S, P, ω, u, π), a competitive equilibrium consists of an allocation c S,P and a price vector p S,P, such that: Agent optimization: For all i I, for each s S, for any vector of consumptions {d i s} i I which is affordable given p s and measurable, we have s 1 P(s) π(s 1 ) π(p(s)) ui (cs(s i 1 )) s 1 P(s) π(s 1 ) π(p(s)) ui (ds(s i 1 )). Measurability: For all s, s S, if P(s) = P(s ), then p s = p s and c i s = c i s for all i I. Feasibility: For each s S, for each s 1 P(s), cs(s i 1 ) ω(s 1 ). i I The Under Unawareness

Single-agent Comparison with HMS Unawareness does not imply that anything goes Proposition (Green (1981)) Suppose that P 1 is uninformative given S and let P 2 be any other possibility correspondence. Then, for some agent, the attained level of ex ante expected utility must be weakly lower in the equilibrium with P 2, than with P 1. Moreover, if his utility function is strictly concave and his consumption in the equilibrium with P 2 is not constant conditional on each payoff relevant state, then the attained level of ex ante expected utility must be strictly lower. The Under Unawareness

Single-agent Comparison with HMS Theorem 2 Suppose that P 2 is more informative than P 1 (P 1, π) and (P 2, π) satisfy Conditional Independence The Under Unawareness

Single-agent Comparison with HMS Theorem 2 Suppose that P 2 is more informative than P 1 (P 1, π) and (P 2, π) satisfy Conditional Independence Then, all agents are weakly worse off under P 2 than under P 1 if either of the following is true: 1. All agents are risk averse and there is no aggregate uncertainty, or, 2. There are enough risk neutral agents who can fully insure the risk averse agents The Under Unawareness

Single-agent Comparison with HMS Projections Preserve Posteriors We do not assume the Projections Preserve Knowledge property of HMS. Its probabilistic analogue, assumed in Heifetz et al. (2013) is the following Definition (P i, π) satisfies Projections Preserve Posteriors if, for all s Σ, if S S S, s S and S i (s) = S, then for any event E S, π(e P i (s S )) = π(e P i (s)). If both s and its projection to S, s S, describe that the agent is aware of event E, then both s and s S specify the same posterior beliefs about E The Under Unawareness

PPP implies CI Single-agent Comparison with HMS Proposition If (P i, π) satisfies Projections Preserve Posteriors, then it satisfies Conditional Independence given S, for each S S. Therefore, the value of information is always positive in the single-agent case and negative in the risk sharing environment The Under Unawareness

Single-agent Comparison with HMS Conclusions Contrary to what is true without unawareness, the value of information can be negative in a single-agent environment and positive in a risk sharing environment The reason for this reversal is that agents treat their awareness signal asymmetrically We provide sufficient conditions that eliminate this reversal and identify two crucial properties of awareness, conditional independence and path-independence The sufficient conditions in the risk sharing environment are stronger than the conditions in the single-agent case The Under Unawareness