This is the eighth post in the Cartesian frames sequence. Here, we define new versions of the sum and tensor of Cartesian frames that can delete spurious possible environments from the frame.
1. Motivating Examples
The sum C⊕D of C and D is supposed to represent an agent that can do anything either C or D can do, while the tensor, C⊗D, is supposed to represent an agent that can do anything C and D can do working freely together on a team. However, sometimes these operations produce Cartesian frames with more environments than we would expect.
Consider two players, Alice and Bob, in a prisoner's dilemma. We will let W={0,1,2,3} be the space of utilities for Alice. The Cartesian frame for Alice looks like
C0=(2031),
where the top row represents Alice cooperating, and the left column represents Bob cooperating.
Let C1=(20) represent Alice committed to cooperating, and let C2=(31) represent Alice committed to defecting. Since the real Alice can either cooperate or defect, one might expect that Alice (C0) would equal the sum (C1⊕C2) of Alice cooperating with Alice defecting. However,
C1⊕C2=(20203113).
The last two columns are spurious environments that represent Bob copying Alice's move, and Bob doing the opposite of Alice's move. However, since Bob cannot see Alice's move, Bob should not be able to implement these policies: Bob can only choose to cooperate or defect.
Next, consider a unilateralist's curse game where two players each have access to a button that destroys the Earth. If either player pushes the button, the Earth is destroyed. Otherwise, the Earth is not destroyed. W={0,1}, where 0 represents the world being destroyed and 1 represents the world not being destroyed.
Here, both players have the Cartesian frame
D1=(0001),
where the first row represents pressing the button, and the first column represents the other player pressing the button.
The two players together can be expressed with the Cartesian frame
D2=⎛⎜
⎜
⎜⎝0001⎞⎟
⎟
⎟⎠,
where the rows in order represent: both players pressing the button; the first player pressing the button; the second player pressing the button; and neither player pressing the button.
One might expect that D1⊗D1 would be D2, but in fact,
D1⊗D1=⎛⎜
⎜
⎜⎝00000010⎞⎟
⎟
⎟⎠.
The second possible environment, in which the earth is just destroyed regardless of what the players do, is spurious.
In both of the above examples, the spurious environments are only spurious because of our interpretation. In the prisoner's dilemma case, C1⊕C2 would be correct if Alice and Bob were playing a modified dilemma where Bob can see Alice's choice. In the unilateralist's curse example, D1⊗D1 would be correct if there were three people playing the game. The problem is that the ⊕ and ⊗ operations do not see our interpretation, and so include all possible environments.
2. Deleting Spurious Environments
We introduce two new concepts, called sub-sum and sub-tensor, which represents sum and tensor with some (but not too many) spurious environments removed.
Definition: Let C=(A,E,⋅), and let D=(B,F,⋆). A sub-sum of C and D is a Cartesian frame of the form (A⊔B,X,⋄), where X⊆Env(C⊕D) and ⋄ is Eval(C⊕D) restricted to (A⊔B)×X, such that C≃(A,X,⋄C) and D≃(B,X,⋄D), where ⋄C is ⋄ restricted to A×X and ⋄D is ⋄ restricted to B×X. Let C⊞D denote the set of all sub-sums of C and D.
Definition: Let C=(A,E,⋅), and let D=(B,F,⋆). A sub-tensor of C and D is a Cartesian frame of the form (A×B,X,∙), where X⊆Env(C⊗D) and ∙ is Eval(C⊗D) restricted to (A×B)×X, such that C≃(A,B×X,∙C) and D≃(B,A×X,∙D), where ∙C and ∙D are given by a∙C(b,x)=(a,b)∙x and b∙D(a,x)=(a,b)∙x. Let C⊠D denote the set of all sub-tensors of C and D.
Thus, we define C⊞D and C⊠D to be sets of Cartesian frames that can be obtained by deleting columns from from C⊕D and C⊗D, respectively, but we have an extra restriction to ensure that we do not delete too many columns.
We will discuss later how to interpret the extra restriction, but first let us go back to our above examples.
If C1=(20) and C2=(31), then C1⊞C2 has 7 elements:
The Cartesian frames above in C1⊞C2 are exactly those with all four entries, 0, 1, 2, and 3. This is because the extra restriction to be in C1⊞C2 is exactly that if you delete the bottom row, you get an object biextensionally equivalent to (20), and if you delete the top row, you get an object biextensionally equivalent to (31).
Similarly, from the unilateralist's curse example,
It is easy to see that there are no other Cartesian frames in D1⊠D1, since there are only four subsets of the two element environment of D1⊗D1, and the Cartesian frames corresponding to the other two subsets do not have 1 in their image, so we cannot build anything biextensionally equivalent to D1 out of them.
Conversely, let C=(A,E,⋅) and D=(B,F,⋆) both be D1, and notice that if
(A×B,X,∙)=⎛⎜
⎜
⎜⎝0001⎞⎟
⎟
⎟⎠,
then (A,B×X,∙C) and (B,A×X,∙D) are both two-by-two matrices with a single 1 entry and three 0 entries, and so must be isomorphic to D1. Similarly, if
(A×B,X,∙)=⎛⎜
⎜
⎜⎝00000010⎞⎟
⎟
⎟⎠,
then (A,B×X,∙C) and (B,A×X,∙D) are both two-by-four matrices with a single 1 entry and seven 0 entries, and so must by biextensionally equivalent to D1.
3. Properties of Sub-Sums and Sub-Tensors
3.1. Sub-Sums and Sub-Tensors Are Commutative
Claim: For any Cartesian frames C0 and C1, there is a bijection between C0⊞C1 and C1⊞C0 that preserves Cartesian frames up to isomorphism. Similarly, there is a bijection between C0⊠C1 and C1⊠C0 that preserves Cartesian frames up to isomorphism.
Proof: Trivial. □
3.2. Tensors Need Not Be Sub-Tensors
For any Cartesian frames C and D with nonempty environments, we have that C⊕D∈C⊞D. However, sometimes C⊗D∉C⊠D. Indeed, sometimes C⊠D={}.
For example, if C and D both have nonempty image, but there are no morphisms from C to D∗, then C⊗D has no environments, and it is easy to see that C⊠D must be empty.
3.3. Sub-Sums and Sub-Tensors Are Superagents
Claim: For any Cartesian frames C0 and C1, and for any D0∈C0⊞C1, we have C0◃D0 and C1◃D0. Similarly, for any D1∈C0⊠C1, we have C0◃D1 and C1◃D1.
Proof: Let Ci=(Ai,Ei,⋅i) and Di=(Bi,Fi,⋆i). First, we show C0◃D0 using the currying definition of subagent. Observe B0=A0⊔A1. Consider the Cartesian frame (A0,{e},⋄) over B0, where ⋄ is given by a⋄e=a. Observe that the definition of sub-sum says that C0≃D∘0(A0,{e},⋄), so C0◃D0. Therefore, C0◃D0, and by commutativity, we also have C1◃D0 .
Similarly, we show C0◃D1 using the currying definition of subagent. Observe that B1=A0×A1. Consider the Cartesian frame (A0,A1,∙) over B1, where ∙ is given by a0∙a1=(a0,a1). Observe that the definition of sub-tensor says that C0≃D∘1(A0,A1,∙). Therefore, C0◃D1, and by commutativity, we also have C1◃D1. □
Observe that in the above proof, the Cartesian frame over B0 we constructed to show that sub-sums are superagents had a singleton environment, and the Cartesian frame over B1 we constructed to show that sub-tensors are superagents had a surjective evaluation function. The relevance of this observation will become clear later.
3.4. Biextensional Equivalence
As we do with most of our definitions, we will now show that sub-sums and sub-tensors are well-defined up to biextensional equivalence.
Claim: Given two Cartesian frames over W, C0 and C1, and given any D∈C0⊞C1, we have that for all C′0≃C0 and C′1≃C1, there exists a D′≃D, with D′∈C′0⊞C′1.
Proof: Let C0=(A0,E0,⋅0), let C1=(A1,E1,⋅1), and let D=(A0⊔A1,X,⋆) be an element of C0⊞C1, so X⊆E0×E1, and a⋆(e0,e1)=a⋅0e0 if a∈A0, and a⋆(e0,e1)=a⋅1e1 if a∈A1.
The fact that D∈C0⊞C1 tells us that for i∈{0,1}, Di≃Ci, where Di=(Ai,X,⋆i) with ⋆i given by a⋆0(e0,e1)=a⋅0e0 and a⋆1(e0,e1)=a⋅0e1.
For i∈{0,1}, let C′i=(A′i,E′i,⋅′i) satisfy C′i≃Ci. Thus, there exist morphisms (gi,hi):Ci→C′i, and (g′i,h′i):C′i→Ci, such that (g0,h0)∘(g′0,h′0), (g′0,h′0)∘(g0,h0), (g1,h1)∘(g′1,h′1), and (g′1,h′1)∘(g1,h1) are all homotopic to the identity.
We define a function f:E0×E1→E′0×E′1 by f(e0,e1)=(h′0(e0),h′1(e1)). Then, we define X′⊆E′0×E′1 by X′={f(x)|x∈X}, and let D′=(A′0⊔A′1,X′,⋆′), where a⋆′(e0,e1)=a⋅′0e0 if a∈A′0, and a⋆′(e0,e1)=a⋅′1e1 if a∈A′1. We need to show D′∈C′0⊞C′1, and that D′≃D.
To show that D′≃D, we will construct a pair of morphisms (j,k):D→D′ and (j′,k′):D′→D that compose to something homotopic to the identity in both orders. We define j:A0⊔A1→A′0⊔A′1 by j(a)=g0(a) if a∈A0, and j(a)=g1(a) if a∈A1. We similarly define j′:A′0⊔A′1→A0⊔A1 by j′(a)=g′0(a) if a∈A′0, and j′(a)=g′1(a) if a∈A′1. We define k′:X→X′ by k′(x)=f(x), which is clearly a function into X′, by the definition of X′. Further, k′ is surjective, and thus has a right inverse. We choose k:X′→X to be any right inverse to k′, so f(k(x))=x for all x∈X′.
To see that (j′,k′) is a morphism, observe that for a∈A′0⊔A′1, and (e0,e1)∈X, if a∈A′i, then
To see D′∈C′0⊞C′1, we need to show that D′i≃C′i, where D′i=(A′i,X′,⋆′i) with ⋆′i given by a⋆′0(e0,e1)=a⋅′0e0 and a⋆′1(e0,e1)=a⋅′1e1. It suffices to show that D′i≃Di, since Di≃Ci≃C′i.
For i∈{0,1}, we construct morphisms (ji,ki):Di→D′i, and (j′i,k′i):D′i→Di. We define ji=gi, j′i=g′i, ki=k, and k′i=k′.
To see that (ji,ki) is a morphism, observe that for all a∈Ai and x∈X′, we have
ji(a)⋆′ix=gi(a)⋆′ix=j(a)⋆′x=a⋆k(x)=a⋆iki(x),
and to see that (j′i,k′i) is a morphism, observe that for all a∈A′i, and x∈X, we have
j′i(a)⋆ix=g′i(a)⋆ix=j′(a)⋆x=a⋆′k′(x)=a⋆′ik′i(x).
To see (j′i,k′i)∘(ji,ki) is homotopic to the identity on Di, observe that for all a∈Ai and x∈X, we have
j′i(ji(a))⋆ix=g′i(gi(a))⋆ix=j′(j(a))⋆x=a⋆x=a⋆ix,
and similarly, to show that (ji,ki)∘(j′i,k′i) is homotopic to the identity on D′i, observe that for all a∈A′i and x∈X′, we have
Thus, we have that D′i≃Di, completing the proof. □
We have a similar result for sub-tensors, whose proof directly mirrors the proof for sub-sums:
Claim: Given two Cartesian frames over W, C0 and C1, and any D∈C0⊠C1, we have that for all C′0≃C0 and C′1≃C1, there exists a D′≃D, with D′∈C′0⊠C′1.
Proof: Let C0=(A0,E0,⋅0), let C1=(A1,E1,⋅1), and let D=(A0×A1,X,⋆) be an element of C0⊠C1, so X⊆hom(C0,C∗1), and
(a,b)⋆(g,h)=a⋅0h(b)=b⋅1g(a).
The fact that D∈C0⊠C1 tells us that for i∈{0,1}, Di≃Ci, where Di=(Ai,A1−i×X,⋆i) with ⋆i given by
a⋆0(b,(g,h))=b⋆1(a,(g,h))=(a,b)⋆(g,h).
For i∈{0,1}, let C′i=(A′i,E′i,⋅′i) satisfy C′i≃Ci. Thus, there exist morphisms (gi,hi):Ci→C′i, and (g′i,h′i):C′i→Ci, such that (g0,h0)∘(g′0,h′0), (g′0,h′0)∘(g0,h0), (g1,h1)∘(g′1,h′1), and (g′1,h′1)∘(g1,h1) are all homotopic to the identity.
We define a function f:hom(C0,C1∗)→hom(C′0,C′1∗) by f(g,h)=(h′1,g′1)∘(g,h)∘(g′0,h′0). This function is well-defined, since (h′1,g′1)=(g′1,h′1)∗∈hom(C∗1,C′1∗) and (h′0,g′0)∈hom(C′0,C0).
Then, we define X′⊆hom(C′0,C′1∗) by X′={f(g,h)|(g,h)∈X}, and let D′=(A′0×A′1,X′,⋆′), where
(a,b)⋆′(g,h)=a⋅′0h(b)=b⋅′1g(a).
We need to show that D′∈C′0⊠C′1, and that D′≃D.
To show that D′≃D, we will construct a pair of morphisms (j,k):D→D′ and (j′,k′):D′→D that compose to something homotopic to the identity in both orders. We define j:A0×A1→A′0×A′1 by j(a,b)=(g0(a),g1(b)), and we similarly define j′:A′0×A′1→A0×A1 by j′(a,b)=(g′0(a),g′1(b)). We define k′:X→X′ by k′(x)=f(x), which is clearly a function into X′, by the definition of X′. Further, k′ is surjective, and thus has a right inverse. We choose k:X′→X to be any right inverse to k′, so f(k(x))=x for all x∈X′.
To see (j′,k′) is a morphism, observe that for (a,b)∈A′0×A′1, and (g,h)∈X, we have
To see D′∈C′0⊠C′1, we need to show that D′i≃C′i, where D′i=(A′i,A′1−i×X′,⋆′i) with ⋆′i given by
a⋆′0(b,(g,h))=b⋆′1(a,(g,h))=(a,b)⋆′(g,h).
It suffices to show that D′i≃Di, since Di≃Ci≃C′i.
For i∈{0,1}, we construct morphisms (ji,ki):Di→D′i, and (j′i,k′i):D′i→Di. We define ji=gi and j′i=g′i. We define ki:(A′1−i×X′)→(A1−i×X) by ki(a,x)=(g′1−i(a),k(x)), and similarly define k′i:(A1−i×X)→(A′1−i×X′) by k′i(a,x)=(g1−i(a),k′(x)).
To see that (j0,k0) is a morphism, observe that for all a∈A0 and (a1,(g,h))∈A′1×X′, we have:
and seeing that (j′1,k′1)∘(j1,k1), (j0,k0)∘(j′0,k′0), and (j1,k1)∘(j′1,k′1) are homotopic to the identity is similar.
Thus, we have that D′i≃Di, completing the proof. □
In our next post, we will use sub-sum and sub-tensor to define additive subagents, which are like agents that have committed to restrict their class of options; and multiplicative subagents, which are like agents that are contained inside other agents. We will also introduce the concept of sub-environments.
This is the eighth post in the Cartesian frames sequence. Here, we define new versions of the sum and tensor of Cartesian frames that can delete spurious possible environments from the frame.
1. Motivating Examples
The sum C⊕D of C and D is supposed to represent an agent that can do anything either C or D can do, while the tensor, C⊗D, is supposed to represent an agent that can do anything C and D can do working freely together on a team. However, sometimes these operations produce Cartesian frames with more environments than we would expect.
Consider two players, Alice and Bob, in a prisoner's dilemma. We will let W={0,1,2,3} be the space of utilities for Alice. The Cartesian frame for Alice looks like
C0=(2031),
where the top row represents Alice cooperating, and the left column represents Bob cooperating.
Let C1=(20) represent Alice committed to cooperating, and let C2=(31) represent Alice committed to defecting. Since the real Alice can either cooperate or defect, one might expect that Alice (C0) would equal the sum (C1⊕C2) of Alice cooperating with Alice defecting. However,
C1⊕C2=(20203113).
The last two columns are spurious environments that represent Bob copying Alice's move, and Bob doing the opposite of Alice's move. However, since Bob cannot see Alice's move, Bob should not be able to implement these policies: Bob can only choose to cooperate or defect.
Next, consider a unilateralist's curse game where two players each have access to a button that destroys the Earth. If either player pushes the button, the Earth is destroyed. Otherwise, the Earth is not destroyed. W={0,1}, where 0 represents the world being destroyed and 1 represents the world not being destroyed.
Here, both players have the Cartesian frame
D1=(0001),
where the first row represents pressing the button, and the first column represents the other player pressing the button.
The two players together can be expressed with the Cartesian frame
D2=⎛⎜ ⎜ ⎜⎝0001⎞⎟ ⎟ ⎟⎠,
where the rows in order represent: both players pressing the button; the first player pressing the button; the second player pressing the button; and neither player pressing the button.
One might expect that D1⊗D1 would be D2, but in fact,
D1⊗D1=⎛⎜ ⎜ ⎜⎝00000010⎞⎟ ⎟ ⎟⎠.
The second possible environment, in which the earth is just destroyed regardless of what the players do, is spurious.
In both of the above examples, the spurious environments are only spurious because of our interpretation. In the prisoner's dilemma case, C1⊕C2 would be correct if Alice and Bob were playing a modified dilemma where Bob can see Alice's choice. In the unilateralist's curse example, D1⊗D1 would be correct if there were three people playing the game. The problem is that the ⊕ and ⊗ operations do not see our interpretation, and so include all possible environments.
2. Deleting Spurious Environments
We introduce two new concepts, called sub-sum and sub-tensor, which represents sum and tensor with some (but not too many) spurious environments removed.
Definition: Let C=(A,E,⋅), and let D=(B,F,⋆). A sub-sum of C and D is a Cartesian frame of the form (A⊔B,X,⋄), where X⊆Env(C⊕D) and ⋄ is Eval(C⊕D) restricted to (A⊔B)×X, such that C≃(A,X,⋄C) and D≃(B,X,⋄D), where ⋄C is ⋄ restricted to A×X and ⋄D is ⋄ restricted to B×X. Let C⊞D denote the set of all sub-sums of C and D.
Definition: Let C=(A,E,⋅), and let D=(B,F,⋆). A sub-tensor of C and D is a Cartesian frame of the form (A×B,X,∙), where X⊆Env(C⊗D) and ∙ is Eval(C⊗D) restricted to (A×B)×X, such that C≃(A,B×X,∙C) and D≃(B,A×X,∙D), where ∙C and ∙D are given by a∙C(b,x)=(a,b)∙x and b∙D(a,x)=(a,b)∙x. Let C⊠D denote the set of all sub-tensors of C and D.
Thus, we define C⊞D and C⊠D to be sets of Cartesian frames that can be obtained by deleting columns from from C⊕D and C⊗D, respectively, but we have an extra restriction to ensure that we do not delete too many columns.
We will discuss later how to interpret the extra restriction, but first let us go back to our above examples.
If C1=(20) and C2=(31), then C1⊞C2 has 7 elements:
(20203113),(202311),(200313),(220313),(020113),(2031),(2013)
The 9 Cartesian frames that can be obtained by deleting columns from C⊕D that are not in C1⊞C2 are:
(2231),(2033),(0211),(0013),(23),(01),(21),(03),().
The Cartesian frames above in C1⊞C2 are exactly those with all four entries, 0, 1, 2, and 3. This is because the extra restriction to be in C1⊞C2 is exactly that if you delete the bottom row, you get an object biextensionally equivalent to (20), and if you delete the top row, you get an object biextensionally equivalent to (31).
Similarly, from the unilateralist's curse example,
D1⊠D1=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜⎝0001⎞⎟ ⎟ ⎟⎠,⎛⎜ ⎜ ⎜⎝00000010⎞⎟ ⎟ ⎟⎠⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭.
It is easy to see that there are no other Cartesian frames in D1⊠D1, since there are only four subsets of the two element environment of D1⊗D1, and the Cartesian frames corresponding to the other two subsets do not have 1 in their image, so we cannot build anything biextensionally equivalent to D1 out of them.
Conversely, let C=(A,E,⋅) and D=(B,F,⋆) both be D1, and notice that if
(A×B,X,∙)=⎛⎜ ⎜ ⎜⎝0001⎞⎟ ⎟ ⎟⎠,
then (A,B×X,∙C) and (B,A×X,∙D) are both two-by-two matrices with a single 1 entry and three 0 entries, and so must be isomorphic to D1. Similarly, if
(A×B,X,∙)=⎛⎜ ⎜ ⎜⎝00000010⎞⎟ ⎟ ⎟⎠,
then (A,B×X,∙C) and (B,A×X,∙D) are both two-by-four matrices with a single 1 entry and seven 0 entries, and so must by biextensionally equivalent to D1.
3. Properties of Sub-Sums and Sub-Tensors
3.1. Sub-Sums and Sub-Tensors Are Commutative
Claim: For any Cartesian frames C0 and C1, there is a bijection between C0⊞C1 and C1⊞C0 that preserves Cartesian frames up to isomorphism. Similarly, there is a bijection between C0⊠C1 and C1⊠C0 that preserves Cartesian frames up to isomorphism.
Proof: Trivial. □
3.2. Tensors Need Not Be Sub-Tensors
For any Cartesian frames C and D with nonempty environments, we have that C⊕D∈C⊞D. However, sometimes C⊗D∉C⊠D. Indeed, sometimes C⊠D={}.
For example, if C and D both have nonempty image, but there are no morphisms from C to D∗, then C⊗D has no environments, and it is easy to see that C⊠D must be empty.
3.3. Sub-Sums and Sub-Tensors Are Superagents
Claim: For any Cartesian frames C0 and C1, and for any D0∈C0⊞C1, we have C0◃D0 and C1◃D0. Similarly, for any D1∈C0⊠C1, we have C0◃D1 and C1◃D1.
Proof: Let Ci=(Ai,Ei,⋅i) and Di=(Bi,Fi,⋆i). First, we show C0◃D0 using the currying definition of subagent. Observe B0=A0⊔A1. Consider the Cartesian frame (A0,{e},⋄) over B0, where ⋄ is given by a⋄e=a. Observe that the definition of sub-sum says that C0≃D∘0(A0,{e},⋄), so C0◃D0. Therefore, C0◃D0, and by commutativity, we also have C1◃D0 .
Similarly, we show C0◃D1 using the currying definition of subagent. Observe that B1=A0×A1. Consider the Cartesian frame (A0,A1,∙) over B1, where ∙ is given by a0∙a1=(a0,a1). Observe that the definition of sub-tensor says that C0≃D∘1(A0,A1,∙). Therefore, C0◃D1, and by commutativity, we also have C1◃D1. □
Observe that in the above proof, the Cartesian frame over B0 we constructed to show that sub-sums are superagents had a singleton environment, and the Cartesian frame over B1 we constructed to show that sub-tensors are superagents had a surjective evaluation function. The relevance of this observation will become clear later.
3.4. Biextensional Equivalence
As we do with most of our definitions, we will now show that sub-sums and sub-tensors are well-defined up to biextensional equivalence.
Claim: Given two Cartesian frames over W, C0 and C1, and given any D∈C0⊞C1, we have that for all C′0≃C0 and C′1≃C1, there exists a D′≃D, with D′∈C′0⊞C′1.
Proof: Let C0=(A0,E0,⋅0), let C1=(A1,E1,⋅1), and let D=(A0⊔A1,X,⋆) be an element of C0⊞C1, so X⊆E0×E1, and a⋆(e0,e1)=a⋅0e0 if a∈A0, and a⋆(e0,e1)=a⋅1e1 if a∈A1.
The fact that D∈C0⊞C1 tells us that for i∈{0,1}, Di≃Ci, where Di=(Ai,X,⋆i) with ⋆i given by a⋆0(e0,e1)=a⋅0e0 and a⋆1(e0,e1)=a⋅0e1.
For i∈{0,1}, let C′i=(A′i,E′i,⋅′i) satisfy C′i≃Ci. Thus, there exist morphisms (gi,hi):Ci→C′i, and (g′i,h′i):C′i→Ci, such that (g0,h0)∘(g′0,h′0), (g′0,h′0)∘(g0,h0), (g1,h1)∘(g′1,h′1), and (g′1,h′1)∘(g1,h1) are all homotopic to the identity.
We define a function f:E0×E1→E′0×E′1 by f(e0,e1)=(h′0(e0),h′1(e1)). Then, we define X′⊆E′0×E′1 by X′={f(x) | x∈X}, and let D′=(A′0⊔A′1,X′,⋆′), where a⋆′(e0,e1)=a⋅′0e0 if a∈A′0, and a⋆′(e0,e1)=a⋅′1e1 if a∈A′1. We need to show D′∈C′0⊞C′1, and that D′≃D.
To show that D′≃D, we will construct a pair of morphisms (j,k):D→D′ and (j′,k′):D′→D that compose to something homotopic to the identity in both orders. We define j:A0⊔A1→A′0⊔A′1 by j(a)=g0(a) if a∈A0, and j(a)=g1(a) if a∈A1. We similarly define j′:A′0⊔A′1→A0⊔A1 by j′(a)=g′0(a) if a∈A′0, and j′(a)=g′1(a) if a∈A′1. We define k′:X→X′ by k′(x)=f(x), which is clearly a function into X′, by the definition of X′. Further, k′ is surjective, and thus has a right inverse. We choose k:X′→X to be any right inverse to k′, so f(k(x))=x for all x∈X′.
To see that (j′,k′) is a morphism, observe that for a∈A′0⊔A′1, and (e0,e1)∈X, if a∈A′i, then
j′(a)⋆(e0,e1)=g′i(a)⋅iei=a⋅′ih′i(ei)=a⋆′(h′0(e0),h′1(e1))=a⋆′k′(e0,e1)).To see that (j,k) is a morphism, consider an arbitrary a∈A0⊔A1 and (e′0,e′1)∈X′, and let (e0,e1)=k(e′0,e′1). Then, if a∈Ai, we have
j(a)⋆′(e′0,e′1)=j(a)⋆′f(e0,e1)=j(a)⋆′(h′0(e0),h′1(e1))=gi(a)⋅′ih′i(ei)=g′i(gi(a))⋅iei=a⋅iei=a⋆(e0,e1)=a⋆k(e′0,e′1).To see that (j′,k′)∘(j,k) is homotopic to the identity on D, observe that for all a∈A0⊔A1 and (e0,e1)∈X, we have that if a∈Ai,
j′(j(a))⋆(e0,e1)=g′i(gi(a))⋅iei=a⋅iei=a⋆(e0,e1).Similarly, to see that (j,k)∘(j′,k′) is homotopic to the identity on D′, observe that for all a∈A′0⊔A′1 and (e0,e1)∈X′, we have that if a∈A′i,
j(j′(a))⋆′(e0,e1)=gi(g′i(a))⋅′iei=a⋅′iei=a⋆′(e0,e1).Thus, D′≃D.
To see D′∈C′0⊞C′1, we need to show that D′i≃C′i, where D′i=(A′i,X′,⋆′i) with ⋆′i given by a⋆′0(e0,e1)=a⋅′0e0 and a⋆′1(e0,e1)=a⋅′1e1. It suffices to show that D′i≃Di, since Di≃Ci≃C′i.
For i∈{0,1}, we construct morphisms (ji,ki):Di→D′i, and (j′i,k′i):D′i→Di. We define ji=gi, j′i=g′i, ki=k, and k′i=k′.
To see that (ji,ki) is a morphism, observe that for all a∈Ai and x∈X′, we have
ji(a)⋆′ix=gi(a)⋆′ix=j(a)⋆′x=a⋆k(x)=a⋆iki(x),and to see that (j′i,k′i) is a morphism, observe that for all a∈A′i, and x∈X, we have
j′i(a)⋆ix=g′i(a)⋆ix=j′(a)⋆x=a⋆′k′(x)=a⋆′ik′i(x).To see (j′i,k′i)∘(ji,ki) is homotopic to the identity on Di, observe that for all a∈Ai and x∈X, we have
j′i(ji(a))⋆ix=g′i(gi(a))⋆ix=j′(j(a))⋆x=a⋆x=a⋆ix,and similarly, to show that (ji,ki)∘(j′i,k′i) is homotopic to the identity on D′i, observe that for all a∈A′i and x∈X′, we have
ji(j′i(a))⋆′ix=gi(g′i(a))⋆′ix=j(j′(a))⋆′x=a⋆′x=a⋆′ix.Thus, we have that D′i≃Di, completing the proof. □
We have a similar result for sub-tensors, whose proof directly mirrors the proof for sub-sums:
Claim: Given two Cartesian frames over W, C0 and C1, and any D∈C0⊠C1, we have that for all C′0≃C0 and C′1≃C1, there exists a D′≃D, with D′∈C′0⊠C′1.
Proof: Let C0=(A0,E0,⋅0), let C1=(A1,E1,⋅1), and let D=(A0×A1,X,⋆) be an element of C0⊠C1, so X⊆hom(C0,C∗1), and
(a,b)⋆(g,h)=a⋅0h(b)=b⋅1g(a).The fact that D∈C0⊠C1 tells us that for i∈{0,1}, Di≃Ci, where Di=(Ai,A1−i×X,⋆i) with ⋆i given by
a⋆0(b,(g,h))=b⋆1(a,(g,h))=(a,b)⋆(g,h).For i∈{0,1}, let C′i=(A′i,E′i,⋅′i) satisfy C′i≃Ci. Thus, there exist morphisms (gi,hi):Ci→C′i, and (g′i,h′i):C′i→Ci, such that (g0,h0)∘(g′0,h′0), (g′0,h′0)∘(g0,h0), (g1,h1)∘(g′1,h′1), and (g′1,h′1)∘(g1,h1) are all homotopic to the identity.
We define a function f:hom(C0,C1∗)→hom(C′0,C′1∗) by f(g,h)=(h′1,g′1)∘(g,h)∘(g′0,h′0). This function is well-defined, since (h′1,g′1)=(g′1,h′1)∗∈hom(C∗1,C′1∗) and (h′0,g′0)∈hom(C′0,C0).
Then, we define X′⊆hom(C′0,C′1∗) by X′={f(g,h) | (g,h)∈X}, and let D′=(A′0×A′1,X′,⋆′), where
(a,b)⋆′(g,h)=a⋅′0h(b)=b⋅′1g(a).We need to show that D′∈C′0⊠C′1, and that D′≃D.
To show that D′≃D, we will construct a pair of morphisms (j,k):D→D′ and (j′,k′):D′→D that compose to something homotopic to the identity in both orders. We define j:A0×A1→A′0×A′1 by j(a,b)=(g0(a),g1(b)), and we similarly define j′:A′0×A′1→A0×A1 by j′(a,b)=(g′0(a),g′1(b)). We define k′:X→X′ by k′(x)=f(x), which is clearly a function into X′, by the definition of X′. Further, k′ is surjective, and thus has a right inverse. We choose k:X′→X to be any right inverse to k′, so f(k(x))=x for all x∈X′.
To see (j′,k′) is a morphism, observe that for (a,b)∈A′0×A′1, and (g,h)∈X, we have
j′(a,b)⋆(g,h)=(g′0(a),g′1(b))⋆(g,h)=g′1(b)⋅1g(g′0(a))=b⋅′1h′1(g(g′0(a)))=(a,b)⋆′(h′1∘g∘g′0,h′0∘h∘g′1)=(a,b)⋆′k′(g,h).To see that (j,k) is a morphism, consider an arbitrary (a,b)∈A0×A1 and (g′,h′)∈X′, and let (g,h)=k(g′,h′). Then, we have:
j(a,b)⋆′(g′,h′)=(g0(a),g1(b))⋆′f(g,h)=(g0(a),g1(b))⋆′(h′1,h′1)∘(g,h)∘(g′0,h′0)=g1(b)⋅′1h′1(g(g′0(g0(a))))=g′1(g1(b))⋅1g(g′0(g0(a)))=b⋅1g(g′0(g0(a)))=g′0(g0(a))⋅0h(b)=a⋅0h(b)=(a,b)⋆(g,h)=(a,b)⋆k(g′,h′).To see that (j′,k′)∘(j,k) is homotopic to the identity on D, observe that for all (a,b)∈A0×A1 and (g,h)∈X, we have:
j′(j(a,b))⋆(g,h)=(g′0(g0(a)),g′1(g1(b)))⋆(g,h)=g′1(g1(b))⋅1g(g′0(g0(a)))=b⋅1g(g′0(g0(a)))=g′0(g0(a))⋅0h(b)=a⋅0h(b)=(a,b)⋆(g,h).Similarly, to see that (j,k)∘(j′,k′) is homotopic to the identity on D′, observe that for all (a,b)∈A′0×A′1 and (g,h)∈X, we have:
j(j′(a,b))⋆′(g,h)=(g0(g′0(a)),g1(g′1(b)))⋆′(g,h)=g1(g′1(b))⋅′1g(g0(g′0(a)))=b⋅′1g(g0(g′0(a)))=g0(g′0(a))⋅′0h(b)=a⋅′0h(b)=(a,b)⋆′(g,h).Thus, D′≃D.
To see D′∈C′0⊠C′1, we need to show that D′i≃C′i, where D′i=(A′i,A′1−i×X′,⋆′i) with ⋆′i given by
a⋆′0(b,(g,h))=b⋆′1(a,(g,h))=(a,b)⋆′(g,h).It suffices to show that D′i≃Di, since Di≃Ci≃C′i.
For i∈{0,1}, we construct morphisms (ji,ki):Di→D′i, and (j′i,k′i):D′i→Di. We define ji=gi and j′i=g′i. We define ki:(A′1−i×X′)→(A1−i×X) by ki(a,x)=(g′1−i(a),k(x)), and similarly define k′i:(A1−i×X)→(A′1−i×X′) by k′i(a,x)=(g1−i(a),k′(x)).
To see that (j0,k0) is a morphism, observe that for all a∈A0 and (a1,(g,h))∈A′1×X′, we have:
a⋆0k0(b,(g,h))=a⋆0(g′1(b),k(g,h))=(a,g′1(b))⋆k(g,h)=j(a,g′1(b))⋆′(g,h)=(g0(a),g1(g′1(b)))⋆′(g,h)=g1(g′1(b))⋅′1g(g0(a))=b⋅′1g(g0(a))=(g0(a),b)⋆′(g,h)=j0(a)⋆′0(b,(g,h)).To see that (j1,k1), (j′0,k′0), and (j′1,k′1) are morphisms is similar.
To see (j′0,k′0)∘(j0,k0) is homotopic to the identity on D0, observe that for all a∈A0 and (b,(g,h))∈A1×X, we have
j′i(ji(a))⋆ix=(g′i(gi(a)),b)⋆(g,h)=g′i(gi(a))⋅0h(b)=a⋅0h(b)=(a,b)⋆(g,h)=a⋆0(b,(g,h)),and seeing that (j′1,k′1)∘(j1,k1), (j0,k0)∘(j′0,k′0), and (j1,k1)∘(j′1,k′1) are homotopic to the identity is similar.
Thus, we have that D′i≃Di, completing the proof. □
In our next post, we will use sub-sum and sub-tensor to define additive subagents, which are like agents that have committed to restrict their class of options; and multiplicative subagents, which are like agents that are contained inside other agents. We will also introduce the concept of sub-environments.