Table 1 Example of the computation of the Owen value
From: A new differential privacy preserving crowdsensing scheme based on the Owen value
Set of coalitions | \( \mathbb{C}=\left\{{\mathcal{C}}_1=\left\{{a}_1,{a}_2\right\},{\mathcal{C}}_2=\left\{{a}_3\right\}\right\} \) | ||
|---|---|---|---|
Permutation | a 1 | a 2 | a 3 |
a1 ← a2 ← a3 | \( \mathcal{V}\left(\left\{{a}_1\right\}\right) \)- \( \mathcal{V}\left(\phi \right) \) =20 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_1\right\}\right) \) = 70 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_1,{a}_2\right\}\right) \) = 30 |
a1 ← a3 ← a2 | N/A; π(a1) < π(a3) < π(a2) and \( {a}_1,{a}_2\in {\mathcal{C}}_1 \) but \( {a}_3\notin {\mathcal{C}}_1 \) | ||
a2 ← a1 ← a3 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_2\right\}\right) \) =60 | \( \mathcal{V}\left(\left\{{a}_2\right\}\right) \)- \( \mathcal{V}\left(\phi \right) \) =30 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_1,{a}_2\right\}\right) \) =30 |
a2 ← a3 ← a1 | N/A; π(a2) < π(a3) < π(a1) and \( {a}_2,{a}_1\in {\mathcal{C}}_1 \) but \( {a}_3\notin {\mathcal{C}}_1 \) | ||
a3 ← a1 ← a2 | \( \mathcal{V}\left(\left\{{a}_1,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_3\right\}\right) \) =40 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_1,{a}_3\right\}\right) \) =40 | \( \mathcal{V}\left(\left\{{a}_3\right\}\right) \)- \( \mathcal{V}\left(\phi \right) \) =40 |
a3 ← a2 ← a1 | \( \mathcal{V}\left(\left\{{a}_1,{a}_2,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_2,{a}_3\right\}\right) \) =50 | \( \mathcal{V}\left(\left\{{a}_2,{a}_3\right\}\right) \)- \( \mathcal{V}\left(\left\{{a}_3\right\}\right) \) =30 | \( \mathcal{V}\left(\left\{{a}_3\right\}\right) \)- \( \mathcal{V}\left(\phi \right) \) =40 |
Total | 20 + 60 + 40 + 50 = 170 | 70 + 30 + 40 + 30 = 170 | 30 + 30 + 40 + 40 = 140 |
Owen value \( {\chi}_a\left(\mathbb{N},{\mathcal{C}}_1,\mathcal{V}\right) \) | 170/4 = 42.5 | 170/4 = 42.5 | 140/4 = 35 |