aiqbench analysis
This article addresses a fundamental question in algorithmic game theory regarding the uniqueness of optimal strategies within formal logical systems, directly engaging with the theoretical underpinnings of AI alignment and cooperative behavior. The discussion centers on Peano arithmetic and the implications for how artificial agents can be programmed to achieve mutual cooperation without external enforcement.
Is there only one FairBot?
The FairBot from the MIRI prisoner's dilemma tournament is defined by a theorem of Peano arithmetic (PA) that holds for each opponent: mjx-container[jax="CHTML"] { line-height: 0; } mjx-container [space="1"] { margin-left: .111em; } mjx-container [space="2"] { margin-left: .167em; } mjx-container [space="3"] { margin-left: .222em; } mjx-container [space="4"] { margin-left: .278em; } mjx-container [space="5"] { margin-left: .333em; } mjx-container [rspace="1"] { margin-right: .111em; } mjx-container [rspace="2"] { margin-right: .167em; } mjx-container [rspace="3"] { margin-right: .222em; } mjx-container [rspace="4"] { margin-right: .278em; } mjx-container [rspace="5"] { margin-right: .333em; } mjx-container [size="s"] { font-size: 70.7%; } mjx-container [size="ss"] { font-size: 50%; } mjx-container [size="Tn"] { font-size: 60%; } mjx-container [size="sm"] { font-size: 85%; } mjx-container [size="lg"] { font-size: 120%; } mjx-container [size="Lg"] { font-size: 144%; } mjx-container [size="LG"] { font-size: 173%; } mjx-container [size="hg"] { font-size: 207%; } mjx-container [size="HG"] { font-size: 249%; } mjx-container [width="full"] { width: 100%; } mjx-box { display: inline-block; } mjx-block { display: block; } mjx-itable { display: inline-table; } mjx-row { display: table-row; } mjx-row > * { display: table-cell; } mjx-mtext { display: inline-block; } mjx-mstyle { display: inline-block; } mjx-merror { display: inline-block; color: red; background-color: yellow; } mjx-mphantom { visibility: hidden; } _::-webkit-full-page-media, _:future, :root mjx-container { will-change: opacity; } mjx-math { display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0; } mjx-container[jax="CHTML"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="CHTML"][display="true"][width="full"] { display: flex; } mjx-container[jax="CHTML"][display="true"] mjx-math { padding: 0; } mjx-container[jax="CHTML"][justify="left"] { text-align: left; } mjx-container[jax="CHTML"][justify="right"] { text-align: right; } mjx-mi { display: inline-block; text-align: left; } mjx-c { display: inline-block; } mjx-utext { display: inline-block; padding: .75em 0 .2em 0; } mjx-mo { display: inline-block; text-align: left; } mjx-stretchy-h { display: inline-table; width: 100%; } mjx-stretchy-h > * { display: table-cell; width: 0; } mjx-stretchy-h > * > mjx-c { display: inline-block; transform: scalex(1.0000001); } mjx-stretchy-h > * > mjx-c::before { display: inline-block; width: initial; } mjx-stretchy-h > mjx-ext { /* IE */ overflow: hidden; /* others */ overflow: clip visible; width: 100%; } mjx-stretchy-h > mjx-ext > mjx-c::before { transform: scalex(500); } mjx-stretchy-h > mjx-ext > mjx-c { width: 0; } mjx-stretchy-h > mjx-beg > mjx-c { margin-right: -.1em; } mjx-stretchy-h > mjx-end > mjx-c { margin-left: -.1em; } mjx-stretchy-v { display: inline-block; } mjx-stretchy-v > * { display: block; } mjx-stretchy-v > mjx-beg { height: 0; } mjx-stretchy-v > mjx-end > mjx-c { display: block; } mjx-stretchy-v > * > mjx-c { transform: scaley(1.0000001); transform-origin: left center; overflow: hidden; } mjx-stretchy-v > mjx-ext { display: block; height: 100%; box-sizing: border-box; border: 0px solid transparent; /* IE */ overflow: hidden; /* others */ overflow: visible clip; } mjx-stretchy-v > mjx-ext > mjx-c::before { width: initial; box-sizing: border-box; } mjx-stretchy-v > mjx-ext > mjx-c { transform: scaleY(500) translateY(.075em); overflow: visible; } mjx-mark { display: inline-block; height: 0px; } mjx-c::before { display: block; width: 0; } .MJX-TEX { font-family: MJXZERO, MJXTEX; } .TEX-B { font-family: MJXZERO, MJXTEX-B; } .TEX-I { font-family: MJXZERO, MJXTEX-I; } .TEX-MI { font-family: MJXZERO, MJXTEX-MI; } .TEX-BI { font-family: MJXZERO, MJXTEX-BI; } .TEX-S1 { font-family: MJXZERO, MJXTEX-S1; } .TEX-S2 { font-family: MJXZERO, MJXTEX-S2; } .TEX-S3 { font-family: MJXZERO, MJXTEX-S3; } .TEX-S4 { font-family: MJXZERO, MJXTEX-S4; } .TEX-A { font-family: MJXZERO, MJXTEX-A; } .TEX-C { font-family: MJXZERO, MJXTEX-C; } .TEX-CB { font-family: MJXZERO, MJXTEX-CB; } .TEX-FR { font-family: MJXZERO, MJXTEX-FR; } .TEX-FRB { font-family: MJXZERO, MJXTEX-FRB; } .TEX-SS { font-family: MJXZERO, MJXTEX-SS; } .TEX-SSB { font-family: MJXZERO, MJXTEX-SSB; } .TEX-SSI { font-family: MJXZERO, MJXTEX-SSI; } .TEX-SC { font-family: MJXZERO, MJXTEX-SC; } .TEX-T { font-family: MJXZERO, MJX
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