The Experiment

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:

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

  2. 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.

Preference weights

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

Calculate No-Choice

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

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

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

  3. 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.

Comparing three algorithms

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

## cardinality threshold (z) =  5 
## Dissimilarity paramter for nest 1 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.0490352 
## Best No-Choice probability value for similarity algorithm =  0.2354007 
## Best No-Choice probability value for balancing algorithm =  0.02249668 
## Best No-Choice probability value for greedy algorithm at point=  10 
## Best No-Choice probability value for similarity algorithm at point=  9 
## Best No-Choice probability value for balancing algorithm at point=  10

## cardinality threshold (z) =  6 
## Dissimilarity paramter for nest 1 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.04859555 
## Best No-Choice probability value for similarity algorithm =  0.2265634 
## Best No-Choice probability value for balancing algorithm =  0.02213086 
## 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 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.0482594 
## Best No-Choice probability value for similarity algorithm =  0.2164265 
## Best No-Choice probability value for balancing algorithm =  0.02187404 
## 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 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.04777829 
## Best No-Choice probability value for similarity algorithm =  0.2085537 
## Best No-Choice probability value for balancing algorithm =  0.02166777 
## Best No-Choice probability value for greedy algorithm at point=  7 
## 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 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.0442147 
## Best No-Choice probability value for similarity algorithm =  0.2027654 
## Best No-Choice probability value for balancing algorithm =  0.01807124 
## 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 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.04124178 
## Best No-Choice probability value for similarity algorithm =  0.1977117 
## Best No-Choice probability value for balancing algorithm =  0.01756818 
## 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 =  0.3 
## Dissimilarity paramter for nest 2 =  0.7 
## Best No-Choice probability value for greedy algorithm =  0.03881456 
## Best No-Choice probability value for similarity algorithm =  0.1934142 
## Best No-Choice probability value for balancing algorithm =  0.01516144 
## 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 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.07245479 
## Best No-Choice probability value for similarity algorithm =  0.1611147 
## Best No-Choice probability value for balancing algorithm =  0.02307882 
## 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) =  6 
## Dissimilarity paramter for nest 1 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.07181931 
## Best No-Choice probability value for similarity algorithm =  0.1598391 
## Best No-Choice probability value for balancing algorithm =  0.02268981 
## 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) =  7 
## Dissimilarity paramter for nest 1 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.0713332 
## Best No-Choice probability value for similarity algorithm =  0.153132 
## Best No-Choice probability value for balancing algorithm =  0.02240895 
## 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 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.06860553 
## Best No-Choice probability value for similarity algorithm =  0.1452411 
## Best No-Choice probability value for balancing algorithm =  0.02097643 
## Best No-Choice probability value for greedy algorithm at point=  7 
## Best No-Choice probability value for similarity algorithm at point=  7 
## Best No-Choice probability value for balancing algorithm at point=  7

## cardinality threshold (z) =  9 
## Dissimilarity paramter for nest 1 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.06426089 
## Best No-Choice probability value for similarity algorithm =  0.1395127 
## Best No-Choice probability value for balancing algorithm =  0.01794484 
## 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 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.06059541 
## Best No-Choice probability value for similarity algorithm =  0.1345643 
## Best No-Choice probability value for balancing algorithm =  0.01700929 
## 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 =  0.4 
## Dissimilarity paramter for nest 2 =  0.6 
## Best No-Choice probability value for greedy algorithm =  0.05757268 
## Best No-Choice probability value for similarity algorithm =  0.1303967 
## Best No-Choice probability value for balancing algorithm =  0.01505388 
## 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 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.1082387 
## Best No-Choice probability value for similarity algorithm =  0.1082387 
## Best No-Choice probability value for balancing algorithm =  0.02317332 
## 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) =  6 
## Dissimilarity paramter for nest 1 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.1073271 
## Best No-Choice probability value for similarity algorithm =  0.1073271 
## Best No-Choice probability value for balancing algorithm =  0.02278627 
## 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) =  7 
## Dissimilarity paramter for nest 1 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.1066293 
## Best No-Choice probability value for similarity algorithm =  0.1066293 
## Best No-Choice probability value for balancing algorithm =  0.02249463 
## 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) =  8 
## Dissimilarity paramter for nest 1 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.09982154 
## Best No-Choice probability value for similarity algorithm =  0.09982154 
## Best No-Choice probability value for balancing algorithm =  0.01992184 
## Best No-Choice probability value for greedy algorithm at point=  7 
## Best No-Choice probability value for similarity algorithm at point=  7 
## Best No-Choice probability value for balancing algorithm at point=  7

## cardinality threshold (z) =  9 
## Dissimilarity paramter for nest 1 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.09468481 
## Best No-Choice probability value for similarity algorithm =  0.09468481 
## Best No-Choice probability value for balancing algorithm =  0.0177765 
## 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 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.09029939 
## Best No-Choice probability value for similarity algorithm =  0.09029939 
## Best No-Choice probability value for balancing algorithm =  0.01627882 
## 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 =  0.5 
## Dissimilarity paramter for nest 2 =  0.5 
## Best No-Choice probability value for greedy algorithm =  0.08664459 
## Best No-Choice probability value for similarity algorithm =  0.08664459 
## Best No-Choice probability value for balancing algorithm =  0.01490722 
## 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 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1611147 
## Best No-Choice probability value for similarity algorithm =  0.07245479 
## Best No-Choice probability value for balancing algorithm =  0.02260181 
## 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) =  6 
## Dissimilarity paramter for nest 1 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1598391 
## Best No-Choice probability value for similarity algorithm =  0.07181931 
## Best No-Choice probability value for balancing algorithm =  0.0222465 
## 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) =  7 
## Dissimilarity paramter for nest 1 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.153132 
## Best No-Choice probability value for similarity algorithm =  0.0713332 
## Best No-Choice probability value for balancing algorithm =  0.02196703 
## Best No-Choice probability value for greedy algorithm at point=  6 
## 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) =  8 
## Dissimilarity paramter for nest 1 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1452411 
## Best No-Choice probability value for similarity algorithm =  0.06860553 
## Best No-Choice probability value for balancing algorithm =  0.01892027 
## Best No-Choice probability value for greedy algorithm at point=  7 
## Best No-Choice probability value for similarity algorithm at point=  7 
## Best No-Choice probability value for balancing algorithm at point=  7

## cardinality threshold (z) =  9 
## Dissimilarity paramter for nest 1 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1395127 
## Best No-Choice probability value for similarity algorithm =  0.06426089 
## Best No-Choice probability value for balancing algorithm =  0.01760974 
## 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 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1345643 
## Best No-Choice probability value for similarity algorithm =  0.06059541 
## Best No-Choice probability value for balancing algorithm =  0.01557972 
## 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 =  0.6 
## Dissimilarity paramter for nest 2 =  0.4 
## Best No-Choice probability value for greedy algorithm =  0.1303967 
## Best No-Choice probability value for similarity algorithm =  0.05757268 
## Best No-Choice probability value for balancing algorithm =  0.01476198 
## 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

## null device 
##           1

Results

The results achieved above indicate the following (when comparing GA and SA):

  1. In the absence of cardinality effect (before z)the greedy algorithm has a steeper slope i.e. it achieves a lower value of no-choice faster.This means that more dissimilar items lead to a more lower value of no-choice when cardinality effect is absent.

  2. In the presence of cardinality (after z)the similarity algorithm rises faster to a higher value of no-choice probability.This means that more similar items lead to a higher value of no-choice under cardinality context-effect.

Both the observations above are consistent with our findings that under NLM more dissimilar items lead to a greater decline in no-choice and under NLM-CO more similar items lead to a greater increase in no-choice.As a result the greedy algorithm always performs better than the similarity algorithm (for all possible cases of lambda 1 and lambda 2).

Our objective is to achieve the lowest possible value of no-choice (under cardinality context effects), therefore we devise the balancing algorithm (BA). Under this algorithm a balance is created between the two extreme cases discussed above (greedy and similarity algorithm). Alternatively, similar items and dissimilar items are added to the nests as assortment depth increases.Since our focus is on cases when cardinality co-exists with similarity consider the first case above when z=3. Under SA, worst performance is achieved for the no-choice probability value. It also leads to a sudden increase in no-choice after z. However, both GA and BA lead to a decrease in no-choice despite cardinality context-effect (because of the dominance of preference weight of added items).Both achieve minima at the same point but BA leads to a better objective function value.

In fact BA will lead to a better value of objective function both with and without choice overload in majority of the cases (Based on the conducted experiments).

END REPORT