Jason Urrutia, Proposal for Intramural 11 vs. 11 Soccer Team Story Section: Our intramural soccer team consists of 15 members, however only 11 may play at a time. The problem is deciding who gets to play. The agents are the players and the captain/manager. The resources are the positions on the field. Players have preferences based on their willingness to participate and skill level. Currently, playing time decisions are dependent on a combination of skill and relationship with the captain, which some team members feel is unfair. Model Section: Agents: 15 players, including one captain/manager. Characteristics include skill level, relationship to the captain, and a desire to play. Preferences: Players want to play in positions that match their ability and to have equal playing time. Allocation is the assignment of players to the eleven playing positions. Mechanism: A set of rules that control allocation according to skill, relationship to the captain, or a combination of the two. e ILetN=1{123..,n} betheset ofteams. Let P = {P1, P2, P3,..., Pn} be the set of positions on the field (resources). Let 1 identify the players, and ci determine their willingness to play/skill level. The allocation of Si(c, E) will aid in determining the playing time and/or position to which a player is assigned, whereas E represents the total amount of playing time or available positions. Axioms: Faimess (Equal Treatment of Equals): e Players with equal claims (based on willingness to engage and ability level) should be given equal playing time or positions. e For any pair of players 1, ] N with equal claims, Si(c, E) = Sj(c, E). e Relevance: This concept assures that players are treated fairly and without partiality, according to their contributions and abilities. Efficiency (Full Compensation for Small Amounts): e Description in English: If the total number of claims for a specific playing position is fewer than or equal to the number of available positions, each player should receive their full claim. Formal Description: If )N min{cj, ci} 0, S(c, E) = S(k c, AE). Relevance: This axiom assures that the system can adjust to changes in scale, such as increased team size or playing positions, without having to be rewritten. These axioms assure fair, efficient, non-manipulable, and flexible allocation of playing time and positions, essential for team harmony. Mechanisms: Proportional Allocation Mechanism: e Allocate playing time or positions in proportion to each player's claim. Players with higher claims get more playing time and better positions. e Formal Description: Si(c, E) determines the allocation for each player i=N. Equal Allocation Mechanism: o Regardless of their claims, all players should be given equal playing time or positions. e The allocation Si(c, E) for each player i 0. Higher Claim Higher Amount For each (c, E), and each i, j E N, if ci 2 cj, then Si(c, E) > Sj(c, E). *Gain G;(c, E, S) = Si(c, E). *Loss Li(c, E, S) = Si(c, E) - ci. *Proportion Pi(c, E, S) = Si(c,E) Equal Treatment of Equals For each (c, E), and each i, j E N, if Ci = Cj, then S;(c, E) = S;(c, E). Scale Invariance For each (c, E), each A E R+, S(c, E) = S(Xc, XE). Anonymity For each (c, E), each permutation o : N - N, and each i E N, Si( c, E) = So(i) ((Co() )iEN, E). Axioms *Minimal Right E - >jemVn Ci- Truncation Doesn't Matter For each (c, E), S(c, E) = S((min {ci, E} )iEN, E). Full Compensation for Small Amounts For each (c, E), if there is i E N such that _jEN min {c;, c;} E, and each i E N, S;(c, E') 2 S;(c, E). My Claim Goes Up I Get More For each (c, E), each i E N, and each c > ci, Si((c;, c_;), E) > Si(c, E). Consistency For each (c, E), each M C N, and each i E N\\ M, S;(c, E) = Si(CM, E - ZiEM Si(c, E))<>