signal_learning_game.ipynb

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    "# Urn model for two agent signal learning game\n",
    "\n",
    "*Adapted from the model described in Argiento et al. (2009) 'Learning to Signal: Analysis of a micro-level reinforcement model'*\n",
    "\n",
    "Two agents -- 'Sender' and 'Receiver' -- are playing a cooperative game, where both get a positive payoff if Receiver's action matches some state in nature (the nature state is determined independently and at random prior to each round). They get no payoff if the action does not match the state. Sender can observe the nature state, and their move consists in deciding which sign to show to Receiver. Receiver can see the sign from Sender (but cannot see the nature state directly), and their choice is which action to take.\n",
    "\n",
    "Given e.g. two relevant states of nature, two signs, and two relevant actions, it is easy to come up with a signalling system which would allow Receiver to match their action to the state of nature. For example, S and R can agree in advance, \"If the state is 1, I will show you A; if it's 2, I will show you B\". Or, equally, they could say \"If 1 then I'll show B, and if 2 then I'll show A\". The challenge is just to decide on *which* code to use, given that either will work equally well.\n",
    "\n",
    "The urn-based model is supposed to show how a stable signalling convention can emerge between two agents *without* this kind of prior communication, and even without them making use of meta-knowledge about their situation. (For instance, if you were playing this game as Receiver with another human, you could probably learn the convention after one round, because you could assume that the Sender will pick one code and stick with it. But signalling systems like this can evolve even in situations where none of the agents can make use of such assumptions, e.g. among birds or bacteria.)\n",
    "\n",
    "This learning model just assumes that each agent has some way of associating what they see (states of nature, or signs) with actions (showing a sign, or taking an action). This is represented here by the idea of each agent having two 'urns' (one for each possible input), which each contain two types of token (one for each output). When the agent receives an input, they draw a token at random from the relevant urn, and the token they draw determines their action. The token is returned at the end of the round. When their combined action is successful (when Receiver's action matches the nature state), they both 'reinforce' the successful behaviour by adding a new token of the same kind to the urn from which they drew.\n",
    "\n",
    "In the game with two states, two signs, and two actions, this simple process of reinforcement will quickly lead the two agent's urns to 'match' in a way that corresponds to one of the solutions sketched above."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3bd17694",
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.random import choice\n",
    "import plotly.graph_objects as go\n",
    "from plotly.subplots import make_subplots\n",
    "from collections import Counter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "10f2f20a",
   "metadata": {},
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   "source": [
    "def proportionOf(arr, element_type):\n",
    "    counts = Counter(arr)\n",
    "    el_count = counts[element_type]\n",
    "    total = sum(counts.values())\n",
    "    return el_count / total"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f21dd88",
   "metadata": {},
   "source": [
    "## Initial Parameters"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "id": "a2bae500",
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   "source": [
    "total_rounds = 200\n",
    "\n",
    "# these don't change; just used as a target for sampling in each round\n",
    "nature_states = ['1', '2']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99af0671",
   "metadata": {},
   "source": [
    "## Run the actual game over `total_rounds` rounds"
   ]
  },
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   "cell_type": "code",
   "execution_count": 22,
   "id": "183c00b4",
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   "source": [
    "# Not essential for running the model, but used to graph\n",
    "# the results.\n",
    "outcome = []\n",
    "sender_proportions_1 = []\n",
    "receiver_proportions_1 = []\n",
    "\n",
    "# The two urns for each agent are stored in a dictionary\n",
    "# with the 'input' labels as keys for ease of updating\n",
    "# these urn objects are updated as learning occurs.\n",
    "# Feel like I should probably use a DataFrame (just store\n",
    "# token counts for each urn, and use them as weights for\n",
    "# selecting between tokens each round?)\n",
    "# but I quite like the simplicity of this approach...\n",
    "sender_urns = { \n",
    "                '1': ['A', 'B'], \n",
    "                '2': ['A', 'B'],\n",
    "            }\n",
    "receiver_urns = { \n",
    "                'A': ['1', '2'], \n",
    "                'B': ['1', '2'] \n",
    "            }\n",
    "\n",
    "for i in range(total_rounds):\n",
    "    state = choice(nature_states)\n",
    "    signal = choice(sender_urns[state])\n",
    "    action = choice(receiver_urns[signal])\n",
    "    sender_proportions_1.append(proportionOf(sender_urns['1'], 'A'))\n",
    "    receiver_proportions_1.append(proportionOf(receiver_urns['A'], '1'))\n",
    "    if action == state:\n",
    "        outcome.append(1)\n",
    "        sender_urns[state].append(signal)\n",
    "        receiver_urns[signal].append(action)\n",
    "    else:\n",
    "        outcome.append(0)"
   ]
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   "source": [
    "## Graphing the outcomes\n",
    "\n",
    "Histogram of cumulative wins, plotted against the proportions of As in Sender Urn 1, and of 1s in Receiver Urn A. (If I've understood it corrently, the hope is that these will correlate over time, both converging towards either 1 or 0.)"
   ]
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{\"anchor\": \"y\", \"domain\": [0.0, 0.94], \"title\": {\"text\": \"Rounds\"}}, \"yaxis\": {\"anchor\": \"x\", \"domain\": [0.0, 1.0], \"showgrid\": false, \"title\": {\"text\": \"Cumulative wins\"}}, \"yaxis2\": {\"anchor\": \"x\", \"overlaying\": \"y\", \"range\": [0, 1], \"showgrid\": false, \"side\": \"right\", \"title\": {\"text\": \"Proportion of tokens in urns\"}}},                        {\"responsive\": true}                    ).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('faacae0a-0466-445e-98e5-f800fcfd4b04');\n",
       "var x = new MutationObserver(function (mutations, observer) {{\n",
       "        var display = window.getComputedStyle(gd).display;\n",
       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
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       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
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       "\n",
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rounds = list(range(0, total_rounds))\n",
    "\n",
    "fig = make_subplots(specs=[[{\"secondary_y\": True}]])\n",
    "fig.add_trace(\n",
    "    go.Histogram(\n",
    "        x=rounds,\n",
    "        y=outcome,\n",
    "        nbinsx=total_rounds,\n",
    "        histfunc=\"sum\",\n",
    "        cumulative={'enabled':True, 'direction': 'increasing', 'currentbin': 'half'},\n",
    "        name=\"Cumulative wins\",\n",
    "        marker={\"color\": \"#b0bec5\"}\n",
    "    ),\n",
    "    secondary_y=False\n",
    ")\n",
    "fig.add_trace(\n",
    "    go.Scatter(\n",
    "        x=rounds,\n",
    "        y=sender_proportions_1,\n",
    "        name=\"Proportion of As in Sender Urn 1\",\n",
    "        marker={\"color\": \"#d81b60\"}\n",
    "    ),\n",
    "    secondary_y=True\n",
    ")\n",
    "fig.add_trace(\n",
    "    go.Scatter(\n",
    "        x=rounds,\n",
    "        y=receiver_proportions_1,\n",
    "        name=\"Proportion of 1s in Receiver Urn A\",\n",
    "        marker={\"color\": \"#3f51b5\"},\n",
    "    ),\n",
    "    secondary_y=True\n",
    ")\n",
    "\n",
    "fig.update_xaxes(title=\"Rounds\")\n",
    "fig.update_yaxes(title=\"Cumulative wins\", showgrid=False, secondary_y=False)\n",
    "fig.update_yaxes(range=[0, 1], title=\"Proportion of tokens in urns\", showgrid=False, secondary_y=True)\n",
    "fig.update_layout(plot_bgcolor=\"#f4f4f4\")\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bd3522e",
   "metadata": {},
   "source": [
    "# Generalising the code for other game structures\n",
    "\n",
    "Game structure fixed by passing in `states` and `signals` arrays. So e.g. you can have a 'two state, three signals' game by passing in `states=['1', '2']` and `signals=['a', 'b']`.\n",
    "\n",
    "Choices are weighted just by urn contents, rather than having any content bias. In theory an initial bias could be introduced either by providing weight constants or by passing in unbalanced urns as starting conditions, e.g. `signals=['a', 'a', 'a', 'b']`. However, neither are possible at present. (Whilst the unbalanced starting-points method would work in the above script, it won't here because tokens are used as column names, which need to be unique.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "bf2aa2a0",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "\n",
    "def signal_simulation(states, signals, total_rounds):\n",
    "    nature_states = states\n",
    "    outcomes = []\n",
    "    \n",
    "    # construct urn dfs\n",
    "    sender_headers = pd.MultiIndex.from_product([states, signals], names=[\"urn\", \"token\"]) \n",
    "    receiver_headers = pd.MultiIndex.from_product([signals, states], names=[\"urn\", \"token\"]) \n",
    "    sender_urns = pd.DataFrame(columns=sender_headers)\n",
    "    receiver_urns = pd.DataFrame(columns=receiver_headers)\n",
    "    \n",
    "    # initialise first row of urns\n",
    "    sender_urns.loc[1] = 1\n",
    "    receiver_urns.loc[1] = 1\n",
    "    \n",
    "    # run model\n",
    "    for i in range(2, total_rounds + 1):\n",
    "        # copy previous round's urns\n",
    "        sender_urns.loc[i] = sender_urns.loc[i - 1]\n",
    "        receiver_urns.loc[i] = receiver_urns.loc[i - 1]\n",
    "        \n",
    "        # choose state of nature at random\n",
    "        state = choice(nature_states)\n",
    "\n",
    "        # choose signal based on state, weighted according to urn contents \n",
    "        sender_numerators = sender_urns.loc[i, state]\n",
    "        sender_denominator = sum(sender_numerators)\n",
    "        sender_weights = [n / sender_denominator for n in sender_numerators]\n",
    "        signal = choice(signals, 1, p=sender_weights)[0]\n",
    "        \n",
    "        # choose action based on signal, weighted according to urn contents\n",
    "        receiver_numerators = receiver_urns.loc[i, signal]\n",
    "        receiver_denominator = sum(receiver_numerators)\n",
    "        receiver_weights = [n / receiver_denominator for n in receiver_numerators]\n",
    "        action = choice(states, 1, p=receiver_weights)[0]\n",
    "        \n",
    "        # update urns if state and action match, and record win/loss outcome\n",
    "        if state == action:\n",
    "            sender_urns.loc[i, (state, signal)] = sender_urns.loc[i, (state, signal)] + 1\n",
    "            receiver_urns.loc[i, (signal, action)] = receiver_urns.loc[i, (signal, action)] + 1\n",
    "            outcomes.append(1)\n",
    "        else:\n",
    "            outcomes.append(0)\n",
    "            \n",
    "    return { 'sender_urns': sender_urns, 'receiver_urns': receiver_urns, 'outcomes': outcomes }"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c8a18df",
   "metadata": {},
   "source": [
    "## Plotting results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "42ad7e48",
   "metadata": {},
   "outputs": [
    {
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       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results = signal_simulation(['1', '2', '3'], ['a', 'b'], 1000)\n",
    "s = results['sender_urns'].copy()\n",
    "\n",
    "fig1 = go.Figure()\n",
    "for column in s.columns:\n",
    "    fig1.add_trace(\n",
    "        go.Scatter(\n",
    "            x=s.index,\n",
    "            y=s[column],\n",
    "            name=\"_\".join(column),\n",
    "        )\n",
    "    )\n",
    "    \n",
    "fig1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "6e234793",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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   ],
   "source": [
    "r = results['receiver_urns'].copy()\n",
    "\n",
    "fig2 = go.Figure()\n",
    "for column in r.columns:\n",
    "    fig2.add_trace(\n",
    "        go.Scatter(\n",
    "            x=r.index,\n",
    "            y=r[column],\n",
    "            name=\"_\".join(column),\n",
    "        )\n",
    "    )\n",
    "    \n",
    "fig2.show()"
   ]
  }
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