This paper studies cost allocation for the bus transportation service in Castellanza, a small town (14,000 inhabitants ca.) close to Varese, Italy. Carlo Cattaneo University (LIUC) is one of the
promoters and funders of this service, together with the City Council and other private agents. The case study is first analysed as a traveling salesman problem (TSP) to find the optimal route. Then the traveling salesman game (TSG) is introduced, where the bus stops are associated with the players of a cooperative game, thus allowing the study of possible allocations of the total cost among them. The optimal route is found by the Branch and Bound algorithm. The Shapley vector and the separable and nonseparable cost are the methods used to allocate the cost of the optimal route among players.

Nicola Besozzi

Luca Ruschetti

Chiara Rossignoli

Fernanda Strozzi

The Traveling Salesman Game for Cost Allocation: The Case Study of the Bus Service in Castellanza

When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the <i>perfectness</i> concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this <i>undominance</i> property however does not extend to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M1" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M2" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 6.92631 6.42315" width="6.92631pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g></svg>-player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.

Slim Belhaiza

On Perfect Nash Equilibria of Polymatrix Games

In gambling scenarios the introduction of taxes may affect playing behavior and the transferred monetary volume. Using a game theoretic approach, we ask the following: How does the transferred monetary volume change when the winner has to pay a tax proportional to her win? In this paper we therefore introduce a new parameter: the <i>expected transfer</i>. For a zerosum matrix game with payoff matrix <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 9.65971 9.04777" width="9.65971pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g></svg> and mixed strategies <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571919pt" id="M2" height="10.6388pt" version="1.1" viewBox="-0.0498162 -7.06688 8.16381 10.6388" width="8.16381pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M3" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg> of the two players it is defined by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.76pt" id="M4" height="15.8527pt" version="1.1" viewBox="-0.0498162 -10.0927 143.235 15.8527" width="143.235pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g190-70" d="M517 162C503 123 484 88 467 68C445 42 417 34 341 34C291 34 256 34 237 47C219 59 213 81 213 128V317H308C395 317 402 311 415 240H444V431H415C403 364 398 356 307 356H213V584C213 613 215 616 246 616H322C394 616 421 609 435 587C448 566 458 544 467 502L496 506C493 557 488 625 488 650H42V622C120 616 128 612 128 523V125C128 43 120 35 29 28V0H511C520 31 540 125 546 158L517 162Z"/></g><g transform="matrix(.0135,0,0,-0.0135,7.724,0)"><path id="g190-85" d="M592 498C586 556 582 629 581 675H560C545 656 537 650 510 650H115C87 650 74 652 60 675H40C38 620 33 555 28 495H57C69 539 77 568 89 585C102 607 122 616 205 616H267V123C267 44 259 34 168 28V0H454V28C360 34 352 44 352 123V616H422C498 616 514 608 531 584C542 568 552 543 562 495L592 498Z"/></g><g transform="matrix(.0135,0,0,-0.0135,16.115,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.802,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.0135,0,0,-0.0135,30.322,0)"><path id="g113-60" d="M114 412C81 412 58 388 58 355C58 321 81 297 113 297S169 321 169 355C169 388 145 412 114 412ZM95 130C70 130 46 114 46 88C46 72 54 65 59 64C93 56 121 33 121 -3C121 -41 93 -68 45 -88L56 -118C117 -99 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,35.682,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.0135,0,0,-0.0135,43.718,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,49.078,0)"><use xlink:href="#g113-114"/></g><g transform="matrix(.0135,0,0,-0.0135,55.73,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,64.203,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,75.941,.007)"><path id="g119-65" d="M697 -46L668 -39C637 -126 611 -142 545 -142H163L474 298L206 694H483C558 694 584 688 600 668C615 651 627 625 639 580L667 585L655 739H70V714L378 254L53 -203V-228H655L697 -46Z"/></g><g transform="matrix(.0135,0,0,-0.0135,88.411,.007)"><use xlink:href="#g119-65"/></g><g transform="matrix(.0135,0,0,-0.0135,100.881,0)"><use xlink:href="#g113-113"/></g><g transform="matrix(.0095,0,0,-0.0095,108.577,3.264)"><path id="g50-106" d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"/></g><g transform="matrix(.0135,0,0,-0.0135,111.74,0)"><use xlink:href="#g113-114"/></g><g transform="matrix(.0095,0,0,-0.0095,118.325,3.264)"><path id="g50-107" d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z"/></g><g transform="matrix(.0135,0,0,-0.0135,122.916,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.0135,0,0,-0.0135,126.479,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.0095,0,0,-0.0095,132.725,3.264)"><use xlink:href="#g50-106"/></g><g transform="matrix(.0095,0,0,-0.0095,134.685,3.264)"><use xlink:href="#g50-107"/></g><g transform="matrix(.0135,0,0,-0.0135,139.287,0)"><use xlink:href="#g113-9"/></g></svg>. Surprisingly, it turns out that for small fair matrix games higher tax rates lead to an <i>increased expected transfer</i>. This phenomenon occurs also in analogous situations with tax on the loser, bonus for the winner, or bonus for the loser. Higher tax or bonus rates lead to overproportional expected revenues for the tax authority or overproportional expected expenses for the grant authority, respectively.

Ingo Alth&#xf6;fer

Marlis B&#xE4;rthel

On Taxed Matrix Games and Changes in the Expected Transfer

Both demand and cost asymmetries are considered in oligopoly model with managerial delegation. It shows that (i) both efficient and inefficient firms with delegation have second move advantage under quantity setting and first move advantage under price competition; (ii) the extended games under both quantity and price competition have subgame equilibria. Lastly, the social welfare of all strategy combinations is considered to find that when the efficient firm moves first and the inefficient firm moves second under price competition, the social welfare can be higher than Bertrand case, if the efficiency gap between the two firms is huge.

Leonard F. S. Wang

Extended Games Played by Managerial Firms with Asymmetric Costs

We study the committee decision making process using game theory. A committee here refers to any group of people who have to select one option from a given set of alternatives under a specified rule. Shenoy (1980) introduced two solution concepts, namely, the one-core and a version of bargaining set for committee games. Shortcomings of these solutions concepts are raised and discussed in this paper. These shortcomings are resolved by introducing two new solutions concepts: the farsighted one-core and the bargaining set revised, inspired by an idea of farsightedness initially defined by Rubinstein (1980). It is shown that the farsighted one-core is always non-empty and is better than the one-core. In a well-specified sense, the bargaining set revised is also better than the bargaining set as defined by Shenoy (1980) and it is always non-empty for simple committee games with linear preferences. Other attractive properties are also proved.

Alphonse Fodouop Fotso

Joseph Armel Momo Kenfack

Bertrand Tchantcho

A Theory of Farsightedness in Committee Games

This paper adds to the discussion, in a general setting, that given a Nash-Schmeidler (nonanonymous) game it is not always possible to define a Mas-Colell  (anonymous)  game. In the two games, the players have different strategic behaviours and the formulations of the two problems are different. Also, we offer a novel explanation for the lack of a Nash equilibrium in an infinite game. We consider this game as the limit of a sequence of approximate, finite games for which an equilibrium exists. However, the limiting pure strategy function is not measurable.

Dionysius Glycopantis

Nash Equilibria in Large Games

We study the dynamic structure of equilibria in game theory. Allowing players in a game the opportunity to renegotiate, or switch to a feasible and Pareto superior equilibrium, can lead to welfare gains. However, in an extensive-form game this can also make it more difficult to enforce punishment strategies, leading to the question of which equilibria are feasible after all. This paper attempts to resolve that question by presenting the first definition of renegotiation-proofness in general games. This new concept, the renegotiation perfect set, satisfies five axioms. The first three axioms&#x2014;namely Rationality, Consistency, and Internal Stability&#x2014;characterize weakly renegotiation-proof sets. There is a natural generalized tournament defined on the class of all WRP sets, and the final two axioms&#x2014;External Stability and Optimality&#x2014;pick a unique &#x201C;winner&#x201D; from this tournament. The tournament solution concept employed, termed the catalog, is based on Dutta&#x2019;s minimal covering set and can be applied to many settings other than renegotiation. It is shown that the renegotiation perfection concept is an extension of the standard renegotiation-proof definition for finite games, introduced by (Benoit and Krishna 1993), and that it captures the notion of a strongly renegotiation-proof equilibrium as defined by (Farrell and Maskin 1989).

Julian C. Jamison

Renegotiation Perfection in Infinite Games

We give recursive definitions for the Banzhaf Value and the Semivalues of cooperative TU games. These definitions were suggested by the concept of potential for the Shapley Value due to Hart and Mas-Colell and by some results of the author who introduced the potentials of these values and the Power Game of a given game.

Irinel Dragan

Some Recursive Definitions for Linear Values of Cooperative TU Games

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Game Theory