These experiments are aimed at exploring choice overload effects (cardinality context effects)when similarity exists (already modeled through the Nested Logit model). The threshold for cardinality (z)is an input paramter and we present results for different values of z. In these experiments we compare the performance of two algorithms:

Greedy Algorithm (GA): Only dissimilar items are added to the assortment

Similarity algorithm(SA): Only similar items added to the assortment

Both are devised to achieve the best possible solution to the assortment optimization problem when choice overload exists under similarity context effects.We consider a fixed set of 10 items to be added to an assortment which is nested into two groups (or nests). The dissimilarity parameter of each of the nest determines the extent to which the items within a nest are similar (or dissimilar to each other.)

For all the experiments below we have a fixed value of alpha =0.02 (magnitude of cardinality effect). However, to obtain the final results we analyze both algorithms for alpha values ranging from 0.01 to 2.

We create a random vector(uniform random variable) for 10 items and sort it(high to low) such that it represents preference weights of items to be added to an assortment

```
## [1] 14.75755 14.51444 14.48424 14.45385 14.40383 13.73213 13.01183
## [8] 11.18438 11.09724 10.56445
```

To calculate no-choice when assortment depth increases, we assume the following:For a given value of the cardinality threshold element ?z? we have:

Fixed value of lambda 1-Disimilarity parameter for nest 1.

Fixed value of lambda 2-Dissimilarity parameter for nest 2.

Fixed number of items in the set I

In addition,we have equal preference weight of both nests when increase in depth begins.This means that before we add an item to the assortment both nests are equally appealing to the customer. Within-nest no-choice exists and is considered to be equal in both nests when the experiment begins. However, with increasing assortment depth, these values will change differently in each nest due to the existing similarity (or dissimilarity).

It is also assumed that the cardinality effect is the same for all experiments (alpha value = 0.02)

A higher value of dissimilarity paramter means that the items are not similar and vice versa for a given nest.

These experiments were performed for multiple values of z,lambda 1 and lambda 2.We perform the experiments here (For a detailed overview on the final results please scroll down to the bottom of the page):

```
## cardinality threshold (z) = 5
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.02455162
## Best No-Choice probability value for similarity algorithm = 0.4095097
## Best No-Choice probability value for balancing algorithm = 0.02294434
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 4
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 6
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.0243337
## Best No-Choice probability value for similarity algorithm = 0.4023842
## Best No-Choice probability value for balancing algorithm = 0.02252711
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 5
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 7
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.0241671
## Best No-Choice probability value for similarity algorithm = 0.39706
## Best No-Choice probability value for balancing algorithm = 0.02219555
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 6
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 8
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.0240481
## Best No-Choice probability value for similarity algorithm = 0.3928888
## Best No-Choice probability value for balancing algorithm = 0.02196989
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 7
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 9
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.02299979
## Best No-Choice probability value for similarity algorithm = 0.3897955
## Best No-Choice probability value for balancing algorithm = 0.01818564
## Best No-Choice probability value for greedy algorithm at point= 8
## Best No-Choice probability value for similarity algorithm at point= 8
## Best No-Choice probability value for balancing algorithm at point= 8
```

```
## cardinality threshold (z) = 10
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.02106576
## Best No-Choice probability value for similarity algorithm = 0.3870726
## Best No-Choice probability value for balancing algorithm = 0.01807347
## Best No-Choice probability value for greedy algorithm at point= 9
## Best No-Choice probability value for similarity algorithm at point= 9
## Best No-Choice probability value for balancing algorithm at point= 9
```

```
## cardinality threshold (z) = 11
## Dissimilarity paramter for nest 1 = 0.1
## Dissimilarity paramter for nest 2 = 0.9
## Best No-Choice probability value for greedy algorithm = 0.01951269
## Best No-Choice probability value for similarity algorithm = 0.3847387
## Best No-Choice probability value for balancing algorithm = 0.01525388
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 10
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 5
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.03403513
## Best No-Choice probability value for similarity algorithm = 0.3182784
## Best No-Choice probability value for balancing algorithm = 0.02195404
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 4
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 6
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.03372868
## Best No-Choice probability value for similarity algorithm = 0.3069326
## Best No-Choice probability value for balancing algorithm = 0.02160988
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 5
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 7
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.03349441
## Best No-Choice probability value for similarity algorithm = 0.2984154
## Best No-Choice probability value for balancing algorithm = 0.02136741
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 6
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 8
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.033327
## Best No-Choice probability value for similarity algorithm = 0.2917302
## Best No-Choice probability value for balancing algorithm = 0.02117319
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 7
## Best No-Choice probability value for balancing algorithm at point= 10
```

```
## cardinality threshold (z) = 9
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.03126263
## Best No-Choice probability value for similarity algorithm = 0.2867709
## Best No-Choice probability value for balancing algorithm = 0.01814666
## Best No-Choice probability value for greedy algorithm at point= 8
## Best No-Choice probability value for similarity algorithm at point= 8
## Best No-Choice probability value for balancing algorithm at point= 8
```

```
## cardinality threshold (z) = 10
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.02887312
## Best No-Choice probability value for similarity algorithm = 0.2824073
## Best No-Choice probability value for balancing algorithm = 0.01790232
## Best No-Choice probability value for greedy algorithm at point= 9
## Best No-Choice probability value for similarity algorithm at point= 9
## Best No-Choice probability value for balancing algorithm at point= 9
```

```
## cardinality threshold (z) = 11
## Dissimilarity paramter for nest 1 = 0.2
## Dissimilarity paramter for nest 2 = 0.8
## Best No-Choice probability value for greedy algorithm = 0.02693967
## Best No-Choice probability value for similarity algorithm = 0.2786703
## Best No-Choice probability value for balancing algorithm = 0.01522324
## Best No-Choice probability value for greedy algorithm at point= 10
## Best No-Choice probability value for similarity algorithm at point= 10
## Best No-Choice probability value for balancing algorithm at point= 10
```

Copyright © 2016 thetazero.com All Rights Reserved. Privacy Policy