Add appendix slides on CGT

src/presentation/defense.qmd

@@ -125,7 +125,7 @@ $\{P, A_1, A_2\}$  | 6
 :::
 
 
-## The Shapley-value
+## The Shapley-value {#sec-cgt-shapley-value}
 
 :::: {.columns}
 
@@ -179,6 +179,13 @@ $\{A_i\}$   | 1
 $\{A_1, A_2\}$  | 2
 $\{P, A_i\}$  | 3
 $\{P, A_1, A_2\}$  | 6
+
+::: {.crossref}
+Core: @sec-cgt-core
+
+Nucleolus: @sec-cgt-nucleolus
+:::
+
 :::
 
 ::::
@@ -1403,6 +1410,135 @@ All chat messages: @sec-chat-topics-all
 :::
 
 
+# Appendix – CGT {visibility="uncounted" .unnumbered .unlisted}
+
+
+## The core {visibility="uncounted" #sec-cgt-core}
+
+:::: {.columns}
+
+::: {.column width="50%"}
+
+There are no profitable deviations
+
+::: {.fragment .smaller}
+$$
+\begin{aligned}
+\pi_P &\geq 0 \\
+\pi_{A_i} &\geq 1 \\
+\pi_{A_1} + \pi_{A_2} &\geq 2 \\
+\pi_{P} + \pi_{A_i} &\geq 3
+\end{aligned}
+$$
+:::
+
+::: {.fragment .smaller}
+May be multi-valued, e.g.:
+
+| $\pi_P$  | $\pi_{A_1}$ | $\pi_{A_2}$ |
+|:--------:|:-----------:|:-----------:|
+|    2     |      2      |      2      |
+|    0     |      3      |      3      |
+|    3     |      2      |      1      |
+:::
+
+:::
+
+::: {.column width="10%"}
+:::
+
+::: {.column width="40%" .smaller}
+
+Players: $$N = \{P, A_1, A_2\}$$
+
+Characteristic function $v(S)$:
+
+Coalition ($S$) | Profits
+----------|---------:
+$\{P\}$     |  0
+$\{A_i\}$   | 1
+$\{A_1, A_2\}$  | 2
+$\{P, A_i\}$  | 3
+$\{P, A_1, A_2\}$  | 6
+:::
+
+::: {.crossref}
+🔙 @sec-cgt-shapley-value
+:::
+
+::::
+
+
+## The nucleolus {visibility="uncounted" #sec-cgt-nucleolus}
+
+:::: {.columns}
+
+::: {.column width="50%"}
+
+Maximize smallest excess
+
+::: {.r-stack}
+
+::: {.fragment .smaller fragment-index=1 .fade-in-then-out}
+Excess: payoff – value
+$$
+\begin{aligned}
+e({P}) &= \pi_P \mathrel{\phantom{=2}} \\
+e({A_i}) &= \pi_{A_i} - 1 \mathrel{\phantom{=1}} \\
+e({P, A_i}) &= \pi_P + \pi_{A_i} - 3 \mathrel{\phantom{=1}} \\
+e({A_1, A_2}) &= \pi_{A_1} + \pi_{A_2} - 2 \mathrel{\phantom{=2}}
+\end{aligned}
+$$
+:::
+
+::: {.fragment .smaller fragment-index=2 .fade-in}
+Excess: payoff – value
+$$
+\begin{aligned}
+e({P}) &= \pi_P \mathrel{\color{RoyalBlue}{=2}} \\
+e({A_i}) &= \pi_{A_i} - 1 \mathrel{\color{RoyalBlue}{=1}} \\
+e({P, A_i}) &= \pi_P + \pi_{A_i} - 3 \mathrel{\color{RoyalBlue}{=1}} \\
+e({A_1, A_2}) &= \pi_{A_1} + \pi_{A_2} - 2 \mathrel{\color{RoyalBlue}{=2}}
+\end{aligned}
+$$
+:::
+
+:::
+
+::: {.fragment .smaller fragment-index=2}
+Unique and contained in the core:
+$$
+\pi_P = \pi_{A_1} = \pi_{A_2} \mathrel{\color{RoyalBlue}= 2}
+$$
+:::
+
+:::
+
+::: {.column width="10%"}
+:::
+
+::: {.column width="40%" .smaller}
+
+Players: $$N = \{P, A_1, A_2\}$$
+
+Characteristic function $v(S)$:
+
+Coalition ($S$) | Profits
+----------|---------:
+$\{P\}$     |  0
+$\{A_i\}$   | 1
+$\{A_1, A_2\}$  | 2
+$\{P, A_i\}$  | 3
+$\{P, A_1, A_2\}$  | 6
+:::
+
+::: {.crossref}
+🔙 @sec-cgt-shapley-value
+:::
+
+::::
+
+
 # Appendix – Chapter 1 {visibility="uncounted" .unnumbered .unlisted}
 
 ## Special case: weighted value {visibility="uncounted" #sec-weighted-value}