On a Unique Solution of the Stochastic Functional Equation Arising in Gambling Theory and Human Learning Process

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left">Event</td><td class="align_center">Probability of occurrence</td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left"><span style="width: 11.5587ptpx;"><svg height="11.6151pt" id="M34" style="vertical-align:-3.18146pt" version="1.1" viewbox="-0.0498162 -8.43364 11.5587 11.6151" width="11.5587pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 173C517 129 499 97 479 75C456 50 421 43 369 43C331 43 297 45 279 53C259 63 255 87 255 135V309H322C407 309 420 305 433 233H469V434H433C420 364 406 359 322 359H255V571C255 604 259 606 296 606H344C400 606 427 599 444 581C462 563 475 530 484 495L521 500C517 562 512 632 511 650H40V615C111 609 121 604 121 517V133C121 48 111 42 26 35V0H536C544 30 564 121 571 166L535 173Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.581,3.132)"><path d="M345 0V33C260 38 258 46 258 122V635C174 612 105 594 35 583V554L87 550C124 546 130 539 130 494V122C130 46 127 38 42 33V0H345Z"></path></g></svg></span></td><td class="align_center"><span style="width: 14.936ptpx;"><svg height="10.2124pt" id="M35" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 14.936 10.2124" width="14.936pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path 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"></path></g><g transform="matrix(.013,0,0,-0.013,7.514,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g></svg></span></td></tr><tr><td class="align_left"><span style="width: 12.6913ptpx;"><svg height="11.6151pt" id="M36" style="vertical-align:-3.18146pt" version="1.1" viewbox="-0.0498162 -8.43364 12.6913 11.6151" width="12.6913pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 173C517 129 499 97 479 75C456 50 421 43 369 43C331 43 297 45 279 53C259 63 255 87 255 135V309H322C407 309 420 305 433 233H469V434H433C420 364 406 359 322 359H255V571C255 604 259 606 296 606H344C400 606 427 599 444 581C462 563 475 530 484 495L521 500C517 562 512 632 511 650H40V615C111 609 121 604 121 517V133C121 48 111 42 26 35V0H536C544 30 564 121 571 166L535 173Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.581,3.132)"><path d="M442 172C405 109 393 108 333 108H171L289 222C383 315 430 374 430 457C430 569 349 635 249 635C187 635 134 610 97 573L34 478L61 455C93 502 136 542 196 542C265 542 300 491 300 422C300 346 261 273 191 190C142 132 84 76 31 25V0H432C443 42 455 105 471 163L442 172Z"></path></g></svg></span></td><td class="align_center"><span style="width: 47.0924ptpx;"><svg height="14.5316pt" id="M37" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 47.0924 14.5316" width="47.0924pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,2.33939e-05)"><path d="M332 -241C217 -138 149 60 149 271C149 483 216 680 332 783L312 808C148 686 74 491 74 272V271C74 50 148 -146 312 -266L332 -241Z"></path></g><g transform="matrix(.013,0,0,-0.013,5.024,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.169,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.706,0)"><path 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"></path></g><g transform="matrix(.013,0,0,-0.013,32.416,2.33939e-05)"><path d="M311 271V272C311 491 237 686 73 808L53 783C169 680 236 483 236 271C236 60 168 -138 53 -241L73 -266C237 -146 311 50 311 271Z"></path></g><g transform="matrix(.013,0,0,-0.013,39.698,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g></svg></span></td></tr><tr><td class="align_left"><span style="width: 12.6ptpx;"><svg height="11.7156pt" id="M38" style="vertical-align:-3.28196pt" version="1.1" viewbox="-0.0498162 -8.43364 12.6 11.7156" width="12.6pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 173C517 129 499 97 479 75C456 50 421 43 369 43C331 43 297 45 279 53C259 63 255 87 255 135V309H322C407 309 420 305 433 233H469V434H433C420 364 406 359 322 359H255V571C255 604 259 606 296 606H344C400 606 427 599 444 581C462 563 475 530 484 495L521 500C517 562 512 632 511 650H40V615C111 609 121 604 121 517V133C121 48 111 42 26 35V0H536C544 30 564 121 571 166L535 173Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.581,3.132)"><path d="M295 385C328 406 353 421 370 436C390 452 402 476 402 502C402 578 343 635 239 635H238C172 635 124 606 97 574L48 506L74 482C102 521 141 558 194 558C239 558 278 530 278 469S219 366 134 339L141 307C156 311 177 313 195 313C246 313 314 274 314 177C314 79 276 43 226 43C166 43 117 89 95 108C82 118 70 116 58 107C45 97 29 80 29 61C29 47 36 34 53 21C73 5 114 -12 164 -12C226 -12 444 54 444 224C444 299 391 365 295 383V385Z"></path></g></svg></span></td><td class="align_center"><span style="width: 46.0746ptpx;"><svg height="12.5351pt" id="M39" style="vertical-align:-3.42946pt" version="1.1" viewbox="-0.0498162 -9.10564 46.0746 12.5351" width="46.0746pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path 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"></path></g><g transform="matrix(.013,0,0,-0.013,9.889,.183)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.404,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.55,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.086,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.013,0,0,-0.013,41.353,.183)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td></tr><tr><td class="align_left"><span style="width: 12.9562ptpx;"><svg height="11.6151pt" id="M40" style="vertical-align:-3.18146pt" version="1.1" viewbox="-0.0498162 -8.43364 12.9562 11.6151" width="12.9562pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M535 173C517 129 499 97 479 75C456 50 421 43 369 43C331 43 297 45 279 53C259 63 255 87 255 135V309H322C407 309 420 305 433 233H469V434H433C420 364 406 359 322 359H255V571C255 604 259 606 296 606H344C400 606 427 599 444 581C462 563 475 530 484 495L521 500C517 562 512 632 511 650H40V615C111 609 121 604 121 517V133C121 48 111 42 26 35V0H536C544 30 564 121 571 166L535 173Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.581,3.132)"><path d="M496 176V231H406V629H337C242 511 133 363 25 211V176H279V113C279 53 274 40 189 33V0H487V33C413 40 406 49 406 109V176H496ZM279 231H103C166 334 228 422 277 488H279V231Z"></path></g></svg></span></td><td class="align_center"><span style="width: 75.8561ptpx;"><svg height="14.5316pt" id="M41" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 75.8561 14.5316" width="75.8561pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,2.33939e-05)"><path d="M332 -241C217 -138 149 60 149 271C149 483 216 680 332 783L312 808C148 686 74 491 74 272V271C74 50 148 -146 312 -266L332 -241Z"></path></g><g transform="matrix(.013,0,0,-0.013,5.024,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.169,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.706,0)"><path 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"></path></g><g transform="matrix(.013,0,0,-0.013,32.416,2.33939e-05)"><path d="M311 271V272C311 491 237 686 73 808L53 783C169 680 236 483 236 271C236 60 168 -138 53 -241L73 -266C237 -146 311 50 311 271Z"></path></g><g transform="matrix(.013,0,0,-0.013,39.698,.183)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.213,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,53.359,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,63.895,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.013,0,0,-0.013,71.163,.183)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Probabilities of the four events.</div>

Journal of Function Spaces

tab4

Table 4

Table 4: On a Unique Solution of the Stochastic Functional Equation Arising in Gambling Theory and Human Learning Process 