# AOS4 - Assignment # 3

A bird-watching aficionado is looking to acquire a new camera. From his favorite guide, she gets information about devices in her price range:

|Model | a | b | c | d |
|---|-----|-----|-----|-----|
| $g_1$ (Picture quality) | 25 | 45 | 90 | 50 |
| $g_2$ (Interface) | 90 | 35 | 20 | 50 |

1. Draw the alternatives in the criteria plane $\mathbb{X}=[0,100]\times[0,100]$
2. Our bird-watcher wants to use the WS model. As she is not super tech-savvy, but does not want to buy crap, she sets equal weight for both criteria. Why is she surprised?
3. Let $g_\lambda := \lambda g_1 + (1-\lambda) g_2$, and $\mathcal G$ the set of all possible functions $g_\lambda$ (here, $\mathcal G \equiv [0,1]$). Plot the four functions $f_x : \lambda \mapsto g_\lambda(x), x\in \{a,b,c,d\}$ for $\lambda\in[0,1]$
4. We define :
* pairwise regret, for a given $g\in \mathcal G$, in and alternatives $x, y \in \mathbb{X}$, by
$$ PR(x,y;g) = g(y)-g(x)$$(*the regret I experience when I choose $y$ instead of $x$*)
* maximum pairwise regret, for alternatives $x, y\in \mathbb{X}$ and set of feasible score functions $\mathcal G$, by
$$ MPR(x,y;\mathcal G) = \max_{g\in \mathcal G} PR(x,y,g)$$
* maximum regret, for alternative $x \in \mathbb X$, subset of alternatives $\mathcal X \subseteq \mathbb X$ and set of feasible score functions $\mathcal G$, by
$$ MR(x, \mathcal X;\mathcal G) = \max_{y\in \mathcal X} MPR(x,y;\mathcal G)$$
* minmax regret, for the sets of feasible alternatives \mathcal X$ and feasible functions $\mathcal G$, by
$$ MMR(\mathcal X ; \mathcal G) = \min _{x\in \mathcal X} MR(x ; \mathcal G)$$

Give interpretations for MPR, MR and MMR. Compute $MMR(\{a,b,c,d\}, \{g_\lambda, \lambda \in [0,1]\})$.
6. The Decision Maker convinces herself that the camera that most suits her need is $d$. Is there a parameter representing this preference?
7. Can this preference be represented in the additive value model? with which marginal utilities?