diff --git a/src/presentation/defense.qmd b/src/presentation/defense.qmd index 0029b64..7640852 100644 --- a/src/presentation/defense.qmd +++ b/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}