Preference Modeling When a DM Must Compare Two Alternatives, S/He Will React in One of the Following Ways:  Indifference (I) a and B Are Same for DM: ‘A I B’

Y. İlker TOPCU, Ph.D.

www.ilkertopcu.net www.ilkertopcu.info www.linkedin.com/in/ilker-topcu twitter.com/yitopcu

facebook.com/yitopcu instagram.com/yitopcu Constructing Decision Model  Assessing performance values of alternatives w.r.t. attributes  Modeling the preferences of DMs  Determining the importance of attributes

www.ilkertopcu.net Preference Modeling When a DM must compare two alternatives, s/he will react in one of the following ways:  Indifference (I) a and b are same for DM: ‘a I b’

 (Strict) Preference (P) Without any hesitation, DM prefers a to b: ‘a P b’

 Weak preference (Q) DM cannot make up her/his mind between ‘a P b’ and ‘a I b’ b is definitely not better than a  ‘a Q b’  Incomparability (R) DM cannot compare a and b (refusal or inability): ‘a R b’ www.ilkertopcu.net Incomparability  Incomparability appears even more frequently when contradictory opinions must be aggregated.  If a decision is finally selected, it may be quite useful, during the decision aid step, to put forward clearly any lack of comparison between actions.

www.ilkertopcu.net Turkish Airlines Euroleague  The competition will be played in three different phases as follows: Regular Season, Playoffs and Final Four  In the regular season, the 18 teams will play in a round-robin format (each team against all the others both home and away) The top eight teams will move on to the next phase, the rest of the teams will be eliminated from the competition.  In the playoffs, that will be held in a best of five games format, the team that wins the series will be the first team to win three games. The winners of the Playoffs will advance to the Final Four, the rest of the teams will be eliminated from the competition.

www.ilkertopcu.net Turkish Airlines Euroleague  In the Final Four, there will be two Semifinals. The losers of the Semifinals will play the Third Place Game. The winners of the Semifinals will play the Championship Game. The winner of the Third Place Game will take the third place. The winner of the Championship Game will be the EuroLeague champions.

 At the end of the season, Champions, Runners-up, Third place, Fourth Place teams are known, they are better than 4 teams eliminated in the playoffs, 4 teams eliminated in the playoffs are incomparable (5th-8th ?), 8 teams eliminated in the regular season are comparable (9th-18th)

www.ilkertopcu.net 2016–17 Turkish Airlines EuroLeague Playoffs

2017 Turkish Airlines EuroLeague Final Four (İstanbul)

Fenerbahçe P Olympiacos P CSKA Moscow P Real Madrid P (Darüşşafaka Doğuş R Panathinaikos R Anadolu Efes R Baskonia) www.ilkertopcu.net FIFA World Cup Final Tournament  In the group stage, The 32 teams compete within eight groups. The top two teams from each group advance to the knockout stage, the rest of the teams will be eliminated: The 16 eliminated teams will be incomparable with each other.  In the knockout stage, the winner of each group will play against runner-up of another group. The winners will qualify to quarter-finals, the losers will be eliminated: The 8 eliminated teams will be incomparable but they will be better than the eliminated teams in the group stage.  In the quarter-finals, the teams will play each other in one-off matches. The winners will qualify to semi-finals, the losers will be eliminated: The 4 eliminated teams will be incomparable but they will be better than the eliminated teams in the previous stages. www.ilkertopcu.net FIFA World Cup Final Tournament  In the semi-finals, the losers of the Semifinals will play the third-place match. the winners of the Semifinals will play the final.  In the third-place match, the winner will be the third team of the tournament, the loser will be the fourth team of the tournament.  In the final, the winner will be the champions of the tournament, the loser will be the runners-up of the tournament.

On 10 January 2017, FIFA approved a new format, the 48-team World Cup (to accommodate more teams), which consists of 16 groups of three teams each, with two teams qualifying from each group, to form a round of 32 knockout stage, to be implemented by 2026

www.ilkertopcu.net Final positions (Russia, 2018)  Champions: FR  Runners-up: HR  Third Place: BE  Fourth Place: EN  Four teams eliminated in the quarter-finals: BR, UY, RU, SE  Eight teams eliminated in the knockout stage: ES, AR, CO, CH, DK, JP, MX, PT  Sixteen teams eliminated in the group stage: PL, SN, DE, AU, TN, PA, KR, RS, CR, NG, IS, PE, IR, MA, SA, EG

France P Crotia P Belgium P England P (Brasil R Uruguay R Russia R Sweden) P (Spain R Argentina R Colombia R Switzerland ...) P (Poland R Senegal R Germany R Australia ...)

www.ilkertopcu.net Traditional Preference Structure  Complete order (P)

ai P aj ⇔ f(ai) > f(aj)

 Complete preorder (P, I)

ai I aj ⇔ f(ai) = f(aj)

www.ilkertopcu.net Turkish Airlines Euroleague Regular Season

 At the end of the Regular Season, a standing will be established based on the number of games won by each team COMPLETE PREORDER (P, I)

 Ties are resolved according to tie breakers  The number of victories in the games played between them.  The goal difference in the games played between them.  All the games played in the current phase of competition firstly by using the overall goal difference and then, if necessary, the greatest number of points scored COMPLETE ORDER (P)

www.ilkertopcu.net Indifference Threshold  Semiorder (P, I, q)

ai P aj ⇔ f(ai) > f(aj) + q

ai I aj ⇔ f(ai) - f(aj) ≤ q  If the difference between performance values of two alternatives is less than the indifference threshold, the DM will be indifferent between these two alternatives.  If the difference is greater than the threshold the DM prefers one to other. DM is indifferent between being in a room at x0C and one at (x+0.001)0C; she simply would not be able to detect the difference. So 200C ~ 20.0010C, 20.0010C ~ 20.0020C, ..., ..., 99.9990C ~ 1000C If DM’s indifferences are transitive, we may chain: 20 0 C ~ 100 0 C. She is equally happy to be just comfortable as to be boiled alive! Therefore we should use a threshold for indifference transitivity. www.ilkertopcu.net Letter Grades

70170513 91.65 AA 70170535 88.09 AA 70160511 86.05 AA 70160517 81.35 BA 70170524 80.55 BA 70170525 80.30 BA 70170505 73.45 BB 70160514 72.80 BB 70160531 72.80 BB 70160525 71.90 BB 70160522 67.20 CB 70160516 66.17 CB  q = 4  Ranking of students w.r.t. letter grades SEMIORDER (P, I, q)  Ranking of students w.r.t. averages COMPLETE PREORDER ( P, I) or COMPLETE ORDER (P)? 72.80 www.ilkertopcu.net Variable Indifference Threshold  Interval order (P, I, q(.))

ai P aj ⇔ f(ai) > f(aj)+q(f(aj))

ai I aj ⇔ f(ai) ≤ f(aj)+q(f(aj)) ∨ f(aJ) ≤ f(ai)+q(f(ai))  Many applications include variations of the indifference threshold along the chosen scale. A variation of $1000 doesn’t mean the same when dealing with thousands of dollars or millions of dollars. “Temperature at a room” vs. “Temperature on the surface of sun.” It is thus often useful to introduce a variable indifference threshold.

www.ilkertopcu.net Indifference & Preference Thresholds

 Pseudo order (P, I, Q, q(.), p(.))

ai P aj ⇔ f(ai) > f(aj)+p(f(aj))

ai Q aj ⇔ f(aj)+p(f(aj)) ≥ f(ai) > f(aj)+q(f(aj))

ai I aj ⇔ f(ai) ≤ f(aj)+q(f(aj)) ∨ f(aJ) ≤ f(ai)+q(f(ai))

www.ilkertopcu.net Indifference & Preference Thresholds  Real life applications show that there is often an intermediary zone inside which the DM hesitates between two different responses or gives contradictory answers depending upon the way questions are put to him.  This observation led to a preference model:  An indifference threshold underneath which the DM shows clear indifference  A preference threshold above which the DM is sure of strict preference

www.ilkertopcu.net Models including Incomparability  Partial order (P, R)  Partial preorder (P, I, R)  Partial semiorder (P, I, R, q)  Partial interval order (P, I, R, q(.))  Partial pseudo order (P, I, R, q(.), p(.))

FIFA World Cup Final tournament results: PARTIAL ORDER (P, R)

Turkish Airlines Euroleague tournament results: PARTIAL ORDER ( P, R)

www.ilkertopcu.net Preference Structures R = ∅ Interval order Partial interval order

Constant threshold Semiorder Partial semiorder

Transitivity of I Complete preorder Partial preorder

No ties Complete order Partial order

www.ilkertopcu.net Attribute’s preference representation

 True Attribute “Traditional model”. If the underlying preference structure is a preorder structure (q=p=0)  Quasi Attribute  Semi Attribute “Threshold model”. If the underlying preference structure is a semiorder structure (q=p>0)  Interval Attribute “Variable threshold model”. If the underlying preference structure is an interval order structure (q(.)=p(.)>0)  Pseudo Attribute “Double threshold model”. If the underlying preference structure is a pseudo order structure (p>q>0)